LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 

Class 


SELF-TAUGHT 

MECHANICAL  DRAWING 

AND  ELEMENTARY 

MACHINE    DESIGN 


A  Treatise 

Comprising  the  First  Principles  of  Geometric  and  Mechanical 
Drawing,  Workshop  Mathematics,  Mechanics,  Strength 
of    Materials,  and  the  Design  of  Machine  Details, 
including   Cams,    Sprockets,    Gearing,    Shafts, 
Pulleys,    Belting,    Couplings,    Screws  and 
Bolts,  Clutches,  Flywheels,  etc.    Pre- 
pared   for  the    Use   of    Practical 
Mechanics  and    Young 
Draftsmen. 

By  F.  L.  .SYLVESTER,  M.E. 

With  Additions 

By  ERIK  OBERG 

Associate  Editor  of  "Machinery,"   Author  of  "Hand-Book  of  Small 

Tools,"  "Shop  Arithmetic  for  the  Machinist,"  "Advanced  Sho^ 

Arithmetic  for  the  Machinist  "  "The  Use  of  Logarithms," 

''Solution  of  Triangles,"  etc. 


FULLY  ILLUSTRATED 


NEW  YORK 

THE    NORMAN    W.    HENLEY    PUBLISHING    CO. 

132  NASSAU  STREET 

1910 


•<   eA 


Copyrighted,  1910,  by 
The  Norman  W.  Henley  Publishing  Co. 


PREFACE 

THE  demand  for  an  elementary  treatise  on 
mechanical  drawing,  including  the  first  principles 
of  machine  design,  and  presented  in  such  a  way 
as  to  meet,  in  particular,  the  needs  of  the  student 
whose  previous  theoretical  knowledge  is  limited, 
has  caused  the  author  to  prepare  the  present  vol- 
ume. It  has  been  the  author's  aim  to  adapt  this 
treatise  to  the  requirements  of  the  practical  me- 
chanic and  young  draftsman,  and  to  present  the 
matter  in  as  clear  and  concise  a  manner  as  possible, 
so  as  to  make  "  self  -study "  easy.  In  order  to  meet 
the  demands  of  this  class  of  students,  practically 
all  the  important  elements  of  machine  design  have 
been  dealt  with,  and,  besides,  algebraic  formulas 
have  been  explained  and  the  elements  of  trigo- 
nometry have  been  treated  in  a  manner  suited  to 
the  needs  of  the  practical  man. 

In  arranging  the  material,  the  author  has  first 
devoted  himself  to  mechanical  drawing,  pure  and 
simple,  because  a  thorough  understanding  of  the 
principles  of  representing  objects  greatly  facilitates 
further  study  of  mechanical  subjects ;  then,  atten- 
tion has  been  given  to  the  mathematics  necessary 

iii 


01  9.J.4.1 


IV  PREFACE 

for  the  solution  of  the  problems  in  machine  design 
presented  later,  and  to  a  practical  introduction  to 
theoretical  mechanics  and  strength  of  materials; 
and,  finally,  the  various  elements  entering  in  ma- 
chine design,  such  as  cams,  gears,  sprocket  wheels, 
cone  pulleys,  bolts,  screws,  couplings,  clutches, 
shafting,  fly-wheels,  etc.,  have  been  treated.  This 
arrangement  makes  it  possible  to  present  a  con- 
tinuous course  of  study  which  is  easily  compre- 
hended and  assimilated  even  by  students  of  limited 
previous  training. 

Portions  of  the  section  on  mechanical  drawing 
was  published  by  the  author  in  The  Patternmaker 
several  years  ago.  These  articles  have,  however, 
been  carefully  revised  to  harmonize  with  the  pres- 
ent treatise,  and  in  some  sections  amplified.  In 
the  preparation  of  the  material,  the  author  has 
also  consulted  the  works  of  various  authors  on 
machine  design,  and  credit  has  been-  given  in  the 
text  wherever  use  has  been  made  of  material  from 
such  sources. 

Several  important  additions  have  been  made  by 
Mr.  Erik  Oberg,  Associate  Editor  of  Machinery. 
In  the  preparation  of  these  additions,  use  has  partly 
been  made  of  material  published  from  time  to  time 
in  Machinery. 

THE  PUBLISHER. 

APRIL,  1910. 


CONTENTS 

PREFACE Page  iii 

CHAPTER   I 

INSTRUMENTS  AND   MATERIALS 

General  Remarks  on  the  Study  of  Drawing — Drawing 
Instruments — Pencils — Use  of  the  Instruments — 
Paper — Ink Page  1 

CHAPTER    II 

DEFINITIONS     OF     TERMS     USED    IN    GEOMETRICAL    AND 
MECHANICAL  DRAWING 

Point — Line  —  Surface —  Solid — Plane— Angle— Circle 
— Parallelogram  —  Polygon —  Ellipse  —  Involute  — 
Cycloid — Parabola Page  10 

CHAPTER   III 

GEOMETRICAL    PROBLEMS 

Bisecting  of  Lines  and  Angles — Perpendicular  Lines — 
Tangents — Regular  Polygons — Inscribed  and  Cir- 
cumscribed Circles — Ellipses — Spirals — Involutes 
— Cycloids — Parabolas Page  17 


Vi  CONTENTS 

CHAPTER   IV 

PROJECTION 

Mode  of  Representing  Objects — Projections  of  Inclined 
Prisms — Surface  Developments  of  Cones  and  Pyra- 
mids— Intersecting  Cylinders,  and  Cylinder  and 
Cone— Projection  of  a  Helix — Isometric  Projec- 
tion  Page  32 

CHAPTER   V 

WORKING  DRAWINGS 

Object  of  Working  Drawings — Assembly  Drawings — 
Detail  Drawings — Dimensions — Finish  Marks — 
Sectional  Views — Cross-section  Chart — Screw 
Threads— Shade  Lines — Tracing  and  Blue-print- 
ing  Page  50 

CHAPTER   VI 

ALGEBRAIC  FORMULAS 

The  Meaning  of  Formulas — Square  and  S'quare  Root — 
Cube  and  Cube  Roots  —  Exponents — Areas  and 
Volumes  of  Plane  Figures  and  Solids Page  79 


CHAPTER  VII 

ELEMENTS  OF  TRIGONOMETRY 

Angles  —  Right-angled  Triangles  —  Trigonometrical 
Functions — Tables  of  Natural  Functions — Solution 
of  Right-angled  Triangles — Solution  of  Oblique- 
angled  Triangles — Laying  Out  Angles  by  Means 
of  Trigonometric  Functions Page  96 


CONTENTS  Vii 

CHAPTER   VIII 
ELEMENTS  OF  MECHANICS 

Resolution  of  Forces—Levers  —  Fixed  and  Movable 
Pulleys — Inclined  Planes — The  Screw — Differential 
Screw — Newton's  Laws  of  Motion — Pendulum — 
Falling  Bodies — Energy  and  Work — Horse-power 
of  Steam  Engines Page  120 


CHAPTER   IX 

FIRST  PRINCIPLES  OF  STRENGTH   OF  MATERIALS 

Factor  of  Safety— Shape  of  Machine  Parts— Strength 
of  Materials  as  Given  by  Kirkaldy's  Tests — Stresses 
in  Castings Page  151 


CHAPTER  X 

CAMS 

General  Principles — Design  of  Cams  Imparting  Uniform 
Motion  —  Reciprocating  Cams  —  Cams  Providing 
Uniform  Return — Uniformly  Accelerated  Motion 
Cams  —  Gravity  Cam  Curve  —  Harmonic  Action 
Cams — Approximate  Gravity  Cam  Curve .  .Page  164 


CHAPTER   XI 

SPROCKET  WHEELS 

Object  of  Sprocket  Wheels  —  Drafting  of  Sprocket 
Wheels  for  Different  Classes  of  Chain— Speed 
Ratio Page  185 


Viii  CONTENTS 

CHAPTER  XII 

GENERAL  PRINCIPLES  OF  GEARING 

Friction  and  Knuckle  Gearing — Epicycloidal  Gearing 
— Gears  with  Strengthened  Flanks — Gears  with 
Radial  Flanks — Involute  Gears — Interference  in 
Involute  Gears — The  Two  Systems  Compared — 
Twenty-degree  Involute  Gears — Shrouded  Gears — 
Bevel  Gears — Worm  Gearing — Circular  Pitch — 
Proportions  of  Teeth  —  Diametral  Pitch  — The 
Hunting  Tooth — Approximate  Shapes  for  Cycloidal 
Gear  Teeth — Involute  Teeth — Proportions  of  Gears 
—Strength  of  Gear  Teeth— Thurston's  Rule  for 
Gear  Shafts— Speed  Ratio  of  Gearing. .  .Page  190 

CHAPTER   XIII 

CALCULATING  THE  DIMENSIONS  OF  GEARS 

Spur  Gearing — Bevel  Gears — Worm  Gearing. .  Page  222 
CHAPTER   XIV 

CONE  PULLEYS 

Conical  Drums  —  Influence  of  Crossed  Belt  —  Cone 
Pulleys  —  Smith's  Rule  for  Laying  Out  Cone 
Pulleys Page  239 

CHAPTER  XV 

BOLTS,  STUDS  AND    SCREWS 

Kinds  of  Screws — United  States  Standard  Screw  Thread 
—Check  or  Lock  Nuts— Bolts  to  Withstand  Shock 
— Wrench  Action — Screws  for  Power  Transmission 
— Efficiency  of  Screws — Acme  Standard  Thread- 
Miscellaneous  Screw  Thread  Systems — Other  Com- 
mercial Forms  of  Screws Page  243 


CONTENTS  ix 

CHAPTER   XVI 

COUPLINGS  AND  CLUTCHES 

Simple  Forms  of  Couplings — Calculation  of  Flange 
Coupling  Bolts  —  Oldham's  Coupling  —  Hooke's 
Coupling  or  Universal  Joint — Toothed  Clutches — 
Friction  Clutches— Cone  Clutches Page  259 


CHAPTER   XVII 

SHAFTS,  BELTS  AND   PULLEYS 

Calculation  of  Shafting— Horse-power  of  Belting — 
Speed  of  Belting — Pulley  Sizes  and  Speed  Ratios 
—Twisted  and  Unusual  Cases  of  Belting .  .Page  272 


CHAPTER    XVIII 

FLY-WHEELS   FOR  PRESSES,  PUNCHES,  ETC. 

Object  of  Fly-wheels — Formulas  for  Fly-wheel  Calcu- 
lations—  Example  of  Fly-wheel  Calculation  for 
Shears Page  289 


CHAPTER  XIX 

TRAINS   OF    MECHANISM 

To  Secure  Increase  of  Speed — To  Secure  Reversal  of 
Direction — The  Compound  Idler — The  Screw  Cut- 
ting Train — Simplified  Rules  for  Calculating  Lathe 
Change  Gears — Back-Gears Page  297 


X  CONTENTS 

CHAPTER   XX 
QUICK   RETURN  MOTIONS 

Object  of  Quick  Return  Motions — Examples  of  Simple 
Designs  of  Quick  Return  Motions — The  Whitworth 
Quick  Return  Device — The  Elliptic  Gear  Quick 
Return Page  313 


SELF-TAUGHT 
MECHANICAL    DRAWING 

CHAPTER  I 

INSTRUMENTS    AND    MATERIALS 

ONE  who  is  to  study  the  subject  of  drawing 
should  not  merely  read  a  book  on  the  subject,  but 
should  prepare  sheets  of  exercises.  This  will  fix 
the  principles  which  he  learns  in  his  mind  in  a  way 
as  reading  alone  will  not  do,  and  will  give  him 
practical  experience  in  the  use  of  the  tools.  The 
geometrical  problems  given  in  this  book  make 
perhaps  the  best  of  subjects  for  a  beginning,  as 
their  proper  execution  will  require  careful  work. 
Later,  the  student  may  make  dimensioned  free- 
hand sketches  of  some  machine  with  which  he  is 
familiar,  and  from  these  sketches  he  may  make  up 
a  set  of  finished  working  drawings.  In  all  of  this 
work,  care  should  be  taken  to  have  it  so  laid  out, 
with  proper  margins  and  spaces  between  different 
parts,  that  the  drawing  when  finished  shall  pre- 
sent an  appearance  of  neatness  and  methodical 
arrangement. 

For  the  purposes  of  the  student,  a  drawing  board 
about  15  by  18  inches  will  be  large  enough.  With 
this  should  be  an  18-inch  T-square,  a  pair  of  6-inch 
triangles,  and  a  set  of  three  or  four  irregular  curves. 

1 


SELF-TAUGHT  MECHANICAL  DRAWING 


For  drawing  full-size  work,  a  good  flat  beveled- 
edge  rule  will  answer  ordinary  requirements,  but 
for  making  half-  or  quarter-size  drawings  some 
kind  of  a  "scale" '  will  be  found  desirable.  The  tri- 
angular scale  shown  in  Fig.  1  is  perhaps  the  one 
mostly  used,  and  it  has  the  advantage  of  possess- 


\  Ytt  \V\\  \  Y\\  \\\\  \\\\  \X\\  \\Tvl\\  \Y\\  \ 


FIG.  1.  -The  Triangular  Scale. 

ing  six  surfaces  for  graduations,  giving  variety 
enough  for  all  sorts  of  conditions,  but  it  has  the 
disadvantage  of  persistently  presenting  the  wrong 
edge,  and  putting  one  to  the  trouble  of  turning  it 
over  and  over  to  get  the  desired  edge.  This  trouble 
may,  of  course,  be  overcome  by  using  a  scale  guard 
such  as  is  shown  in  Fig.  2,  but  the  guard  is  itself 

often  in  the  way.    As 
but  two  or  three  differ- 
ent scales,   aside  from 
full  size,  will  be  likely 
to  be  required,  it  will  be 
found  much  more  con- 
FIG.  2. -Scale  Guard  or  Holder  venient  to  have  a  sep- 
used  on  Triangular  Scale.       arate  flat  scale  for  each 

graduation.    Such  scales 

may  be  purchased,  or,  if  one  is  satisfied  with  the 
open  graduation  system  shown  in  Fig.  3,  he  may 
make  them  without  much  trouble  himself.  In  this 
system,  only  one  inch  is  divided,  this  inch  being 
numbered  0;  and  measurements  which  include  a 


INSTRUMENTS  AND  MATERIALS  3 

fractional  part  of  an  inch  are  reckoned  from  the 
required  whole  number  to  the  proper  place  on  the 
divided  inch. 

The  drawing  instruments  themselves,  while  not 
necessarily  of  the  highest  price,  should  be  of  a 
good  serviceable  quality  of  German  silver.  The 
cheap  brass  or  nickel  plated  school  sets  should  not 
be  considered,  as  they  will  prove  unsatisfactory 
for  regular  work.  It  is  not  necessary  to  have  a 
large  number  of  instruments.  A  very  good  set, 
sufficient  for  all  ordinary  requirements,  might  be 
as  follows:  First  a  pair  of  about  4J-  or  5-inch  com- 


11     12     13     14     15     16    17  "18 


FIG.  3.— Inexpensive  Type  of  Scale. 

passes  with  fixed  needle  points  (bayonet  points  are 
useless)  and  interchangeable  pin  and  pencil  points, 
with  lengthening  bar.  Then,  a  pair  of  hair- 
spring spacers  of  about  the  same  size.  These  re- 
semble ordinary  plain  compasses,  but  the  steel  end 
of  one  leg  is  made  adjustable  by  means  of  a 
thumb  screw.  Next,  a  pair  of  ruling  pens,  one 
large  and  one  small,  and,  lastly,  a  set  of  three 
spring  instruments,  pen,  pencil  and  spacers,  for 
small  work.  Rather  than  to  get  cheap  instru- 
ments, it  would  be  advisable  to  obtain  a  set  gradu- 
ally by  getting  the  large  instruments  and  one  pen 
first,  and  adding  the  second  pen  and  the  spring 
instruments  later.  The  large  compasses  can,  if 
necessary,  be  used  to  make  circles  of  from  about 
i  inch  to  about  18  or  20  inches  in  diameter,  so 


4  SELF-TAUGHT  MECHANICAL  DRAWING 

that  they  will  do  very  well  for  a  beginning.  For 
making  larger  circles,  beam  compasses,  in  which 
separate  heads  for  the  needle  point  and  for  the  pen 
or  pencil  point  are  attached  to  a  wooden  bar,  after 
the  manner  of  workmen's  trammels,  are  used. 

A  convenient  case  for  the  instruments,  when 
they  are  bought  separately,  is  shown  in  Fig.  4, 
and  is  made  as  follows:  Take  two  pieces  of 
chamois  skin  or  thin  broadcloth,  one  of  them  about 
one-half  longer  than  the  longest  instrument,  and 
somewhat  wider  than  all  of  them  when  they  are 


FIG.  4.— Home-made  Instrument  Case. 

laid  out  side  by  side,  and  the  second  one  of  the 
same  width  as  the  first,  but  somewhat  shorter  than 
the  longest  instrument.  This  second  piece  is 
sewed  onto  the  large  piece  at  one  end  by  the  outer 
edges.  Pockets  for  the  reception  of  the  instru- 
ments are  then  made  as  shown,  and  when  the  free 
end  of  the  large  piece  is  folded  over,  the  instru- 
ments are  rolled  up  together. 

The  pencils,  which  to  avoid  scratching  particles, 
should  be  of  best  quality,  should  not  be  sharpened 
to  a  round  point,  but  to  a  flat  oval  point,  as  such 
a  shape  will  wear  longer  than  a  round  point ;  the 
leads  used  in  the  compasses,  however,  should  be 


INSTRUMENTS  AND  MATERIALS  5 

only  slightly  flattened.  It  will  be  found  desirable 
to  have  two  grades  of  pencils,  one  quite  hard, 
about  "4H,"  to  be  used  for  laying  out  work,  and 
a  softer  one,  about  "2H,"  to  be  used  for  going 
over  the  lines  of  work  which  is  not  to  be  inked  in. 
In  laying-out  work  where  the  hard  pencil  is  used, 
only  a  moderate  pressure  should  be  applied,  so  as 
to  permit  of  erasures  at  any  time,  whether  for  the 
purpose  of  making  alterations,  or  to  free  the  draw- 
ing of  pencil  marks  after  inking. 

The  drawing  pens  should  be  kept  sharp,  though 
not  so  sharp  as  to  cut  the  paper,  and  their  ends 
should  present  a  neat  oval  shape.  The  needle 
points  of  the  compasses  should  also  be  kept  sharp 
to  avoid  the  tendency  to  slip  when  doing  work 
where  it  is  undesirable  to  prick  through  the  pa- 
per. A  small  Arkansas  stone  will  be  found  useful 
for  this  purpose.  Where  much  use  is  made  of  a 
given  center,  it  may  be  desirable  to  employ  a  horn 
or  metal  center,  such  as  are  kept  in  stock  by  deal- 
ers in  artists'  supplies,  to  avoid  the  troublesome 
enlargement  of  the  center  in  the  paper  which  the 
points  of  the  compasses  would  otherwise  make. 

In  making  a  drawing,  care  should  be  taken  to 
have  the  preliminary  pencil  work  done  correctly. 
It  is  a  mistake  which  beginners  are  likely  to  make, 
to  think  that  errors  in  the  pencil  work  may  be 
readily  corrected  in  the  inking.  This,  however, 
is  usually  another  case  where  "haste  makes 
waste."  It  is  much  better  to  spend  a  little  extra 
time  on  the  pencil  work,  than  to  have  to  throw 
away  a  nearly  finished  ink  drawing  and  do  the 
work  all  over  again.  In  locating  the  various 


6 


SELF-TAUGHT  MECHANICAL  DRAWING 


views  of  a  drawing  upon  the  paper,  it  will  fre- 
quently be  found  to  be  well  to  make  rough  sketches 
of  it  on  scrap  paper.  These  sketches  can  then  be 
moved  around  on  the  drawing  paper  until  the  best 
arrangement  is  secured. 

In  making  a  drawing,  it  will  be  found  most  con- 
venient, ordinarily,  to  limit  the  use  of  the  T-square 
to  horizontal  lines,  the  head  of  the  square  being 
kept  pressed  firmly  against  the  left-hand  end  of 
the  drawing  board.  Vertical  lines  are  then  made 


FIG.  5.  —Appearance  of  Carelessly  made  Drawing. 

with  the  aid  of  the  triangles  resting  against  the 
blade  of  the  T-square.  Vertical  lines  which  are 
too  long  to  be  made  in  this  way,  are,  of  course, 
made  with  the  T-square  itself.  In  inking  in  a 
drawing,  it  is  best  to  draw  all  curved  or  circular 
lines  first,  as  it  is  easier  to  join  straight  lines  onto 
curved  lines  than  to  join  curved  lines  onto  straight 
lines.  Care  should  also  be  taken  to  have  meeting 
lines  just  meet,  whether  they  meet  end  to  end  or 
at  an  angle.  Carelessness  in  this  respect  gives  a 
drawing  a  very  bad  appearance,  as  shown  by  Fig. 
5,  A  and  B. 


INSTRUMENTS  AND  MATERIALS  7 

In  using  the  pens,  whether  the  ruling  or  the  com- 
pass pens,  care  should  be  taken  to  see  that  both  nibs 
rest  upon  the  paper,  otherwise  lines  such  as  shown 
in  Fig.  6  may  result.  If  the  pen  does  rest  squarely 
upon  the  paper,  and  such  lines  continue  to  appear, 
it  is  fair  to  infer  that  the  paper  has  become  some- 
what greasy,  perhaps  from  too  much  handling. 
This  trouble  may  be  avoided,  and  the  work  kept 
cleaner,  by  having  a  piece  of  thin  paper  inter- 
posed between  the  hands  and  the  drawing  paper. 

The  cross  hatching  work,  such  as  is  shown  at  A 
in  Fig.  5,  is  frequently  done  by  simply  using  one 
of  the  triangles  resting  against  the  blade  of  the 


FlG.  6.— Line  Resulting  from  not  Having  both  Pen  Points  or 
Nibs  Resting  on  the  Paper  when  Inking. 

T-square,  the  same  as  is  done  for  vertical  lines,  the 
spacing  being  done  entirely  by  the  eye;  but  unless 
one  is  doing  a  good  deal  of  this  work,  so  as  to 
keep  in  practice,  he  will  find  it  very  difficult  to 
make  the  spacing  regular.  There  are  various  sec- 
tion-lining devices  on  the  market  for  doing  this 
work,  some  of  them  quite  expensive.  Fig.  7  shows 
a  simple  device  for  cross-sectioning,  which  serves 
the  purpose  as  well  as  any  of  the  more  elaborate 
ones,  and  possesses  the  additional  advantage  that 
anyone  may  readily  make  it  for  himself.  This 
instrument  was  shown  by  Mr.  E.  W.  Beardsley  in 
Machinery,  September,  1905.  An  old  instrument 
screw,  B,  is  screwed  into  a  slightly  smaller  hole 
in  a  piece  of  wood,  A,  shaped  as  shown,  and  of  a 


8 


SELF-TAUGHT  MECHANICAL  DRAWING 


thickness  a  little  in  excess  of  the  diameter  of  .the 
screw-head.  This  combination  is  then  used  in  the 
central  hole  in  a  triangle,  as  shown.  Then,  with 
one  finger  on  the  triangle  itself,  and  with  another 
one  on  A,  the  two  may  be  moved  along,  first  one 
and  then  the  other,  for  section  lining,  the  desired 
width  of  space  being  secured  by  the  adjustment 
given  to  B. 

For  making  erasures  of  ink  lines  on  paper,  a 
steel  scraping  eraser  or  a  sharp  knife  blade  is  usu- 


FIG.  7.  —  Simple  Cross-section  Liner. 

ally  the  best,  the  roughened  surface  being  after- 
wards rubbed  down  smooth  with  some  hard  sub- 
stance. When  making  erasures  of  either  pencil 
or  ink  with  a  rubber  eraser,  an  erasing  shield, 
such  as  is  shown  in  Fig.  8,  is  useful  for  prevent- 
ing rubbing  out  more  than  is  intended.  These 
shields  are  made  both  of  thin  sheet  metal  and  of 
celluloid;  the  metal  ones,  being  the  thinner,  are 
the  more  convenient  to  use. 
Tho  paper  used,  if  good  work  is  desired,  should 


INSTRUMENTS  AND  MATERIALS  9 

be  regular  drawing  paper,  whether  it  be  white  or 
brown.  This  has  an  unglazed  surface,  and  will  be 
found  much  more  satisfactory  in  every  way  than 
common  paper.  The  glazed  surface  of  the  cheaper 
paper  does  not  take  pencil  marks  well,  and  is  torn 
up  badly  in  making  erasures.  Such  paper,  if  used 
at  all,  should  be  used  only  on  the  most  temporary 


FIG.  8.— Erasing  Shield  made  from  Sheet  Metal 
or  Celluloid. 


work.  Of  white  drawing  papers,  the  smooth  sur- 
faced kinds  should  be  selected.  For  making  ink 
drawings,  it  will  be  found  most  satisfactory  to  use 
the  prepared  drawing  inks,  rather  than  to  go  to 
the  trouble  of  preparing  it  oneself  from  the  stick 
India  ink. 

For  fastening  the  paper  on  to  the  board,  common 
one-half-ounce  copper  tacks  are  as  good,  if  not 
preferable,  to  other  fastening  means. 


CHAPTER    II 

DEFINITIONS     OF     TERMS     USED     IN     GEOMETRICAL 
AND    MECHANICAL    DRAWING 

1.  A  Point  has  position,  but  not  magnitude. 

2.  A  Line  has  length,  but  neither  breadth  nor 
thickness. 

3.  A  Surface  has  length  and  breadth,  but  not 
thickness. 

4.  A  Solid  has  length,  breadth  and  thickness. 

5.  A  Plane  is  a  surface  which   is  straight  in 
every  direction;  that  is,  one  which  is  perfectly 
flat. 

6.  Parallel  lines  are  such  as  are  everywhere 
equally  distant  from  each  other.     Circular  lines 
which  answer  to  this  condition  are  also  said  to  be 
concentric. 

7.  An  Angle  is  the  difference  in  the  direction 
of  two  lines.     If  the  lines  meet,  the  point  of  meet- 
ing is  called  the  vertex  of  the  angle,  and  the  lines 
ab  and  ac,  Fig.  9,  are  its  sides. 

8.  If  a  straight  line  meets  another  so  that  the 
adjacent  angles  are  equal,  each  of  these  angles  is 
a  right  angle,  and  the  two  lines  are  perpendicular 
to  each  other.    Thus  the  angles  acd  and  deb,  Fig. 
10,  are  right  angles,  and  the  lines  ab  and  dc  are 
perpendicular  to  each  other.     A  distinction  is  to 
be  made  here  between  the  words  perpendicular 

10 


DEFINITIONS  OF  TERMS 


11 


and  vertical.  A  vertical  line  is  one  which  is  per- 
pendicular to  the  plane  of  the  earth's  horizon ;  that 
is,  to  the  surface  of  still  water. 

9.  An  Obtuse  Angle   is  one  which  is  greater 
than  a  right  angle,  as  ace,  Fig.  10. 

10.  An  Acute  Angle  is  one  which  is  less  than  a 
right  angle,  as  ecb,  Fig.  10. 

11.  It  is  obvious  that  the  sum  of  all  the  angles 
which  may  be  formed  about  the  point  c,  Fig.  10, 
above  the  line  ab  will  be  equal  to  the  two  right 
angles  acd  and  deb. 


FIG.  9.— Angle. 


FIG.  10.  —Illustration  for  Making 
Clear  the  Terms  Right,  Acute 
and  Obtuse  Angles. 


12.  The  Complement  of  an  angle  is  a  right  angle, 
less  the  given  angle.     Thus  bcet  Fig.  10,  is  the 
complement  of  dee. 

13.  The  Supplement  of  an  angle  is  two  right 
angles  less  the  given  angle.     Thus  bee,  Fig.  10,  is 
the  supplement  of  ace. 

14.  A  Circle  is  a  continuous  curved  line,  Fig.  11, 
or  the  space  enclosed  by  such  line,  every  point  of 
which  is  equally  distant  from  a  point  within  called 
the  center. 

15.  The    distance    across    a    circle,    measured 
through  the  center,  is  the  diameter.     The  distance 
around  the  circle  is  the  circumference.     The  dis- 


12  SELF-TAUGHT  MECHANICAL  DRAWING 

tance  from  the  center  to  the  circumference  is  the 
radius. 

16.  The  ratio  between  the  circumference  and 
the  diameter,  that  is,  the  circumference  divided 
by  the  diameter,  is  3.1416.    While  this  is  not  exact 
(Bradbury's  Geometry  states  that  it  has  been  car- 
ried out  to  two  hundred  and  fifty  places  of  deci- 
mals), it  is  near  enough  for  practical  purposes. 
This  ratio  is  frequently  represented  by  the  Greek 
letter  TT  (pi). 

17.  A  circle  is  considered  as  being  equally  divided 


FIG.    11. -Illustration    for        FIG.  12. -Similar  Triangles. 
Making  Clear  the  Terms 
Relating  to  the  Circle. 

into  three  hundred  and  sixty  degrees  (360°),  each 
degree  into  sixty  minutes  (60'),  and  each  minute 
into  sixty  seconds  (60"). 

18.  If  two  diameters  cross  each  other  at  right 
angles,  the  circle  is  divided  into  four  equal  parts; 
hence  a  right  angle  contains  ninety  degrees. 

19.  An  Arc  of  a  circle  is  any  part  of  its  circum- 
ference, as  abc,  Fig.  11. 

20.  A  Chord  is  a  straight  line  joining  the  ends 
of  an  arc,  as  ac,  Fig.  11. 

21.  Two  triangles,  as  abc  and  dec,  Fig.  12,  hav- 
ing like  angles  are  similar  triangles.     The  corre- 


DEFINITIONS  OF  TERMS 


13 


spending  sides  of  similar  triangles  have  the  same 
ratio.  Thus  if  ac  were  twice  as  long  as  dc,  ab 
would  be  twice  as  long  as  de,  and  be  would  be 
twice  as  long  as  ec. 

22.  The  sum  of  the  angles  of  a  triangle  is  equal 
to  two  right  angles.  Let  abc,  Fig.  13,  represent 
any  triangle.  Extend  one  side,  ac,  as  shown,  and 
make  cd  parallel  with  ab.  Then  the  angle  dee  is 
equal  to  the  angle  bac,  for  their  sides  have  the 
same  direction,  and  the  angle  bed  is  equal  to  the 


FIG.  13.— Illustration  for 
Showing  that  the  Sum  of 
the  Angles  in  a  Triangle 
equals  Two  Right  Angles. 


FIG.  14.— Tangent   and  Nor- 
mal to  a  Curve. 


angle  abc,  for  their  sides  have  opposite  directions; 
hence  the  sum  of  the  three  angles  formed  about 
the  point  c  is  equal  to  the  sum  of  the  three  angles 
of  the  triangle  abc,  and  these  are  equal  to  two 
right  angles  (11). 

23.  A  Tangent  is  a  line  which  touches  another, 
but  does  not,  though  extended,  cross  it.  Thus,  a, 
b  and  c,  Fig.  14,  are  tangent  lines.  A  line,  d, 
perpendicular  to  the  straight  line  6,  at  the  point 
of  tangency,  is  called  a  normal.  If  one  of  the 


14 


SELF-TAUGHT  MECHANICAL  DRAWING 


lines,  as  a,  is  circular,  the  normal  will  pass  through 
its  center. 

24.  A  Parallelogram  is  a  figure  whose  opposite 
sides  are  parallel,  as  ab  and  cd,  or  eb  and  fd  in 
Fig.  15.  The  sides  may  all  be  of  equal  length, 


FIG.  15.  —Parallelograms. 


FIG.  16. -Square. 


(See 


when  the  parallelogram  is  called  a  square. 
Fig.  16.) 

25.  Figures  having  five,  six  or  eight  sides  are 
called  respectively  Pentagon,  Hexagon  and  Octagon. 
These,  and  all  figures  having  more  than  four  sides, 
are  called  Polygons.  If  the  sides  in  a  polygon  are 


FIG.  17.— Regular  Polygon. 


FIG.  18.— Ellipse. 


all  of  equal  length,  and  all  the  angles  equal,  the 
polygon  is  called  a  regular  polygon.  (See  Fig.  17.) 
26.  An  Ellipse,  Fig.  18,  is  a  continuous  curved 
line,  or  the  space  enclosed  by  such  line,  of  such 
shape  that  the  sum  of  the  distances  from  two 


DEFINITIONS  OF  TERMS 


15 


points  within,  as  a  and  6,  called  the  foci  (singu- 
lar: focus),  to  any  point  upon  its  circumference 
is  constant.     Thus  al  plus  bl  equals  a2  plus  b2  or 
a3  plus  b3. 
27.  An  Involute  is  a  line  of  such  shape  (as  a  in 


FIG.  19.— Involute. 


FIG.  20.— Cycloid. 


Fig.  19)  as  might  be  made  by  a  pencil  at  the  end 
of  a  string  which  is  unwound  from  a  circle. 

28.  A  Cycloid  is  a  line  of  such  shape  (as  a  in 
Fig.  20)  as  might  be  made 

by  a  pencil  fastened  to  the 
circumference  of  a  circle 
which  is  being  rolled  upon 
a  straight  line.  If  the  circle 
was  being  rolled  upon  the 
convex  side  of  a  circular 
line  the  line  traced  by  the 
pencil  would  be  an  epicy- 
cloid. If  it  was  being  rolled 
upon  the  concave  side  of  a 
circular  line,  the  line  traced 
by  the  pencil  would  be  a 
hypocycloid.  The  involute 
and  cycloidal  curves  are  used  in  gear  outlines. 

29.  A  Parabola  is  a  curve  which  may  be  ob- 


FIG.  21.  Method  of  Sec- 
tioning a  Cone  to  Ob- 
tain a  Parabola. 


16  SELF-TAUGHT  MECHANICAL  DRAWING 

tained  by  cutting  a  cone  so  that  the  exposed 
sectional  surface  will  be  parallel  with  one  of  the 
sides  of  the  cone,  as  shown  in  Fig.  21.  This 
curve,  as  shown  in  Fig.  22,  is  of  such  shape  that 
lines  drawn  to  it  from  a  certain  point  within, 
called  the  focus,  shown  at  /  in  the  illustration, 
niake  the  same  angle  with  it  as  lines  drawn  from 


/7 


/\7 


FIG.  22.     Parabola. 

the  intersection  points  parallel  with  the  axis  ax. 
Thus  the  line  fm  makes  the  same  angle  with  the 
parabola,  at  the  point  of  intersection,  as  the  line 
ml.  Because  of  this  property  of  the  parabola, 
mirrors  of  this  shape  are  used  in  headlights  of 
locomotives,  in  search  lights,"  and  in  many  light- 
houses ;  because,  if  a  light  be  placed  at  the  focus, 
its  rays,  when  reflected  from  the  mirror,  will  be 
thrown  out  in  parallel  lines. 


CHAPTER  III 

GEOMETRICAL    PROBLEMS 

Prob.  1,  Fig.  23.  To  bisect  a  line,  either  curved 
as  abc,  or  straight  as  ac. — With  centers  at  a  and  c 
and  with  a  radius  somewhat  greater  than  half  the 
length  of  the  line,  describe  the  arcs  d  and  e.  A 
line  passing  through  the  intersections  of  these  arcs 
bisects  either  line.  It  will  also  pass  through  the 
center  of  the  circle  of  which  the  arc  abc  is  a  part. 

Prob.  2,    Fig.   24.     To   bisect   an  angle. — With 


FIG.  23. — Bisecting  a  Line.        FIG.  24.— Bisecting  an  Angle. 

center  at  a,  and  with  any  convenient  radius,  de- 
scribe the  arc  be.  With  centers  at  b  and  c,  and 
with  a  radius  greater  than  half  the  arc,  describe 
the  arcs  d  and  e.  A  line  from  a  through  the  inter- 
section of  these  arcs  bisects  the  angle. 

Prob.  3,  Fig.  25.  To  make  an  angle  equal  to  a 
given  angle. — Let  a  be  the  given  angle,  and  let  it 
be  desired  to  make  an  angle  equal  to  it  on  the  line 
dg.  With  center  at  a  make  the  arc  be,  and  then 
with  center  at  d  make  the  arc  eh  with  the  same 

17 


18 


SELF-TAUGHT  MECHANICAL  DRAWING 


radius.  Then  with  a  radius  equal  to  be,  and  with 
center  at  h,  make  the  arc  /.  A  line  from  d  through 
the  intersection  of  the  arcs  gives  the  required 
angle. 

Prob.  4,  Fig.  26.     To  erect  a  perpendicular  at  the 
end  of  a  line,  ab. — With  any  convenient  center,  c, 


FIG.  25. — Making  an  Angle  Equal  to  a  Given  Angle. 

and  with  radius  cb,  draw  a  semicircle  intersecting 
ab  at  d.  Draw  a  line  from  d  through  c  intersect- 
ing the  semicircle  at  e.  A  line  from  6  passing 
through  e  is  the  required  perpendicular. 

Prob.  5,  Fig.  27.     To  drop  a  perpendicular  from 
a  point  a,  to  a  given  line  be. — With  a  as  a  center, 


FIG.  26.— Erecting  a  Perpen- 
dicular Line. 


r 

FIG.  27. — Drawing  a  Perpen- 
dicular Line. 


draw  an  arc  intersecting  be  at  d  and  e.  With  d 
and  e  as  centers  draw  the  intersecting  arcs  /  and 
g.  A  line  from  a  through  the  intersection  of 
these  arcs  is  the  required  perpendicular.  If  a 
were  over  one  end  of  the  line  be  the  process  shown 


GEOMETRICAL  PROBLEMS  19 

in  the  preceding  problem  might  be  reversed  by 
drawing  a  line  from  a  corresponding  to  de,  Fig. 
26,  and  upon  this  line  drawing  a  semicircle,  when 
its  intersection  with  the  base  line  would  give  the 
point  to  which  the  perpendicular  from  a  should  be 
drawn. 

Prob.  6,  Fig.  28.  To  draw  a  tangent  to  a  circle 
at  a  given  point. — Draw  a  radius  of  the  circle  to 
the  required  point,  and  erect  a  perpendicular  to  it, 
which  will  be  the  required  tangent.  To  find  the 
point  of  tangency  of  a  line  to  a  circle,  drop  a  per- 


FIG.  28.— Drawing  a  Tangent      FIG.  29.— Finding  the  Center 
to  a  Circle.  of  a  Circle. 

pendicular  to  the  tangent  from  the  center  of  the 
circle. 

Prob.  7,  Fig.  29.  To  find  the  center  of  a  circle.— 
Mark  off  two  arcs  as  ab  and  ac  upon  the  circumfer- 
ence, and  bisect  these  arcs  as  in  Prob.  1.  Where 
these  bisecting  lines  cross  each  other  will  be  the 
required  center. 

Prob.  8,  Fig.  30.  To  draw  a  regular  hexagon 
upon  a  given  base,  ab. — With  a  radius  equal  to  the 
length  of  ab  draw  the  arcs  c  and  d.  The  intersec- 
tion of  these  arcs  will  be  the  center  of  a  circum- 
scribing circle  upon  which  the  other  sides  may  be 
marked  off. 


20 


SELF-TAUGHT  MECHANICAL  DRAWING 


Prob.  9,  Fig.  SI.  To  draw  a  regular  octagon  in 
a  square. — Draw  the  diagonals  of  the  square,  ad 
and  be,  and  with  a  radius  equal  to  half  of  a  diago- 
nal, and  with  centers  at  a,  b,  c  and  d,  draw  the 
arcs  e,  f,  g  and  h.  The  intersections  of  these  arcs 


FIG.  30.— Drawing  a  Regular 
Hexagon. 


FIG.  31. — Drawing  a  Regular 
Octagon. 


with  the  sides  of  the  square  give  the  corners  of 
the  required  octagon. 

Prob.  10,  Fig.  32.  To  draw  a  circle  about  a  tri- 
angle, as  abc. — Bisect  any  two  of  the  sides  as  in 
Prob.  1.  Where  the  bisecting  lines  cross  each 


32. — Drawing  a  Circle 
about  a  Triangle. 


FIG.  33.— Inscribing  a  Circle 
in  a  Triangle. 


other  will  be  the  center  of  the  required  circle.  In 
a  similar  manner  a  center  may  be  found  from 
which  to  draw  a  circle  through  any  three  given 
points,  the  given  points  in  this  case  being  the  cor- 
ners of  the  triangle. 


GEOMETRICAL  PROBLEMS 


21 


Prob.  11,  Fig.  33.  To  draw  a  circle  within  a 
given  triangle,  as  abc. — Bisect  any  two  of  the  angles 
as  in  Prob.  2.  Where  the  bisecting  lines  cross,  will 
be  the  center  of  the  required  circle.  In  a  similar 
manner  a  center  may  be  found  from  which  to  draw 
a  circle  tangent  to  any  three  given  straight  lines. 

Prob.  12,  Fig.  34.  To  find  the  foci  of  an  ellipse.  — 
Draw  the  long  and  the  short  diameters  of  the 
ellipse,  ab  and  cd,  and  with  a  radius  equal  to  half 
of  the  long  diameter,  and  with  a  center  at  c  or  d 


FIG.  34.— Finding  the  Foci  of 
an  Ellipse. 


FIG.  35.— Simplified  Method 
of  Drawing  an  Ellipse. 


draw  the  arcs  e  and  /.  Where  these  arcs  intersect 
the  long  diameter  will  be  the  required  foci. 

Prob.  13,  Fig.  35.  To  draw  an  ellipse  with  a 
pencil  and  thread. — Having  found  the  foci  of  the 
ellipse,  stick  a  pin  firmly  into  each  focus,  and  loop- 
ing a  thread  around  them,  allow  it  to  be  slack 
enough  so  that  the  pencil  will  draw  it  out  to  the 
end  of  the  short  diameter.  The  thread  will  then 
guide  the  pencil  so  that  it  will  draw  an  ellipse.  A 
groove  should  be  cut  around  the  pencil  lead  to  pre- 
vent the  thread  from  slipping  off. 

Prob.  14,  Fig.  36.  To  draw  an  ellipse  with  a 
trammel. — Lay  out  the  long  and  the  short  diame- 
ters of  the  ellipse,  ab  and  cd,  and  on  a  strip  of 
paper,  A,  mark  off  1-3  equal  to  half  of  the  long  diam- 


22  SELF-TAUGHT  MECHANICAL  DRAWING 

eter,  and  2-3  equal  to  half  of  the  short  diameter. 
Then,  keeping  point  1  on  the  short  diameter,  and 
point  2  on  the  long  diameter,  mark  off  any  desired 
number  of  points  at  3.  A  curved  line  passing 
through  these  points  will  be  the  required  ellipse. 
The  ellipsograph,  an  instrument  for  drawing  el- 
lipses, is  made  on  this  principle,  points  at  1  and 
2  traveling  in  grooves  which  coincide  with  ab 
and  cd. 

Prob.  15,  Fig.  37.     To  draw  an  ellipse  by  tangent 
lines. — Make  ab  equal  to  one-half  of  the  long  di- 


d 


FIG.  36.— Another  Method  of      FIG.  37.— Drawing  an  Ellipse 
Drawing  an  Ellipse.  by  Tangents. 

ameter  of  the  required  ellipse,  and  be  equal  to  one- 
half  its  short  diameter.  Divide  ab  and  be  into 
the  same  number  of  equal  parts,  and,  numbering 
them  as  indicated,  connect  1  and  1' ',  2  and  2'  and 
so  forth.  A  curved  line  starting  at  a,  tangent  to 
these  lines,  and  ending  at  c,  is  one-quarter  of  the 
required  ellipse. 

Prob.  16,  Fig.  38.  To  draw  an  approximate  el- 
lipse with  compasses,  using  four  centers. — Lay  out 
the  long  diameter  ab,  and  the  short  diameter  cd, 
crossing  each  other  centrally  at  o.  From  6  meas- 
ure off  be  equal  to  co,  one-half  of  the  short  diam- 
eter. The  length  ae  will  then  be  the  radius  gh 
for  forming  the  part  hk  of  the  ellipse.  From  e 


GEOMETRICAL  PROBLEMS 


23 


mark  off  the  point/,  making  ef  equal  to  one  half 
of  oe.  The  point  /  will  be  the  center,  and  fb  the 
radius  for  forming  the  end  of  the  ellipse.  Lines 
drawn  from  the  centers  g  through  the  points  /  de- 
termine the  points  at  which  the  different  curves 
meet.  This  method  is  not  considered  applicable 
when  the  short  diameter  is  less  than  two-thirds  of 
the  long  diameter. 


FIG.  38. — Drawing  an  Approximate  Ellipse  by  Four 
Circular  Arcs. 

Prob.  17,  Figs.  39  and  39a.  To  draw  an  approx- 
imate ellipse  with  compasses,  using  eight  centers.— 
Lay  out  the  long  diameter  ab,  and  the  short  diam- 
eter cd  crossing  each  other  centrally  at  /.  Con- 
struct the  parallelogram  aecf,  and  draw  the  diago- 
nal ac.  From  e  draw  a  line  at  right  angles  to  ac, 
crossing  the  long  diameter  at  h,  and  meeting  the 
short  diameter,  extended,  at  g.  Point  g  is  the  center 
from  which  to  strike  the  sides  of  the  ellipse,  and 


24  SELF-TAUGHT  MECHANICAL  DRAWING 

h  will  be  the  center,  subject  to  certain  modifica- 
tions for  narrow  ellipses,  from  which  to  strike  the 
ends  of  the  ellipse.  To  get  the  radius  of  the  third 
curve  for  connecting  the  side  and  end  curves,  lay 
off  a  base  line  ab,  Fig.  39 A,  of  any  convenient 
length,  and  divide  it  into  five  equal  parts  by  the 
points  1,  2,  3  and  4.  At  one  end  of  the  line  erect 
the  perpendicular  ac,  equal  to  the  end  radius  ah, 
and  at  the  other  end  erect  the  perpendicular  bd 
equal  to  the  side  radius  eg.  Connect  the  ends  of 
these  perpendiculars  by  the  line  cd,  and  at  point 
2  erect  a  perpendicular,  meeting  cd  at  e.  The 
length  e2  will  be  the  desired  third  radius.  With 
the  compasses  set  to  this  radius,  find  a  center  i 
from  which  a  curve  can  be  struck  which  will  be 
just  tangent  to  the  side  and  end  curves.  From 
other  centers  similarly  located  the  remainder  of 
the  ellipse  is  drawn.  Lines  drawn  from  i  through 
h,  and  from  g  through  i  determine  the  meeting 
points  of  the  different  curves. 

For  narrow  ellipses  the  length  of  the  end  radius, 
ah,  should  be  increased  as  follows :  For  an  ellipse 
having  its  breadth  equal  to  one-half  of  its  length, 
make  ah  one-eighth  longer.  For  an  ellipse  having 
its  breadth  one-third  of  its  length,  make  ah  one- 
fourth  longer.  For  an  ellipse  having  its  breadth 
equal  one-quarter  of  its  length,  make  ah  one-half 
longer.  For  intermediate  breadths  lengthen  ah 
proportionately.  With  this  modification  of  the 
length  of  the  end  radius,  this  method  gives  curves 
which  blend  well  together  so  as  to  satisfy  the  eye, 
and  gives  a  figure  which  conforms  quite  closely  to 
the  actual  outlines  of  an  ellipse. 


GEOMETRICAL  PROBLEMS 


25 


FIG.  39a. 

FIGS.  39  and  39a. — Drawing  an  Approximate  Ellipse  by 
Eight  Circular  Arcs. 


26  SELF-TAUGHT  MECHANICAL  DRAWING 

Prob.  18,  Fig.  40.  To  draw  a  regular  polygon  of 
any  number  of  sides  on  a  given  base,  ab.—  Extend  ab 
as  shown,  and  on  it  with  one  end  as  a  center  and 
a  radius  equal  to  the  length  of  the  given  side,  draw 
a  semicircle.  Divide  this  semicircle  into  as  many 
equal  spaces  as  there  are  to  be  sides  to  the  polygon. 
A  line  from  b  to  the  second  space,  reckoning  from 
where  the  semicircle  meets  the  extension  of  ab, 
will  be  a  second  side  of  the  required  polygon. 
Lines  are  then  drawn  from  b  through  the  remain- 
ing divisions  of  the  semicircle,  and  the  remaining 


FIG.  40. — Drawing  a  Regular      FIG.  41. — Drawing   a   Spiral 
Pentagon.  about  a  Square. 

sides  of  the  polygon  are  marked  off  upon  them  as 
indicated.  If  the  polygon  is  to  have  many  sides, 
as  an  additional  precaution  against  error,  bisect  ab 
and  b2,  thus  getting  the  center  of  a  circumscribing 
circle  upon  which  the  remaining  sides  may  be 
marked  off. 

Prob.  19,  Fig.  41.  To  draw  a  spiral  about  a 
square. — Lay  out  a  square,  1-2-3-4,  having  the 
length  of  each  side  equal  to  one-quarter  of  the  de- 
sired distance  between  the  successive  convolutions 
of  the  spiral,  and  extend  each  side  in  one  direction 
as  shown.  With  a  center  at  2,  and  with  a  radius 
1-2  draw  a  quarter  of  a  circle.  With  a  center  at  3 


GEOMETRICAL  PROBLEMS  27 

draw  another  quarter  of  a  circle,  continuing  the 
first  one,  and  so  continue  with  successive  corners 
of  the  square  for  centers. 

Fig.  42  shows  how,  by  similarly  extending  one 
end  of  each  side,  a  spiral  may  be  drawn  about  a 
regular  polygon  of  any  number  of  sides.  A  curve 
so  formed  determines  the  shape  of  the  teeth  of 
sprocket  wheels. 

Prob.  20,  Fig.  43.  To  draw  an  involute. — Upon 
the  circumference  of  the  given  circle  mark  off  any 


FIG.  42. — Drawing   a   Spiral      FIG.  43.— Drawing  an   Invo- 
about  a  Regular  Polygon.  lute. 

number  of  equally  distant  points,  as  0-1-2-3,  etc., 
and  draw  lines  tangent  to  the  circle  at  these  points, 
beginning  at  point  1.  Then  with  the  compasses 
set  the  same  as  for  marking  off  the  spaces  on  the 
circle,  mark  off  one  space  on  line  1,  two  spaces  on 
line  2,  three  spaces  on  line  3t  and  so  forth.  A 
curved  line  starting  at  0  and  passing  through  these 
points  will  be  the  required  involute.  This  curve 
is  used  for  the  shape  of  the  teeth  of  involute  gears. 
Prob.  21,  Fig.  44.  To  draw  a  cycloid. — Upon  the 
base  line  ab  mark  off  any  number  of  equally  dis- 
tant points,  as  0-1-2-3,  etc.,  the  distance  between 


28 


SELF-TAUGHT  MECHANICAL  DRAWING 


them  being  made,  for  convenience  sake,  about  one- 
sixth  of  half  the  circumference  of  the  generating 
circle.  Beginning  at  1  erect  perpendiculars  from 
these  points,  and  with  centers  on  these  lines  draw 
arcs  of  circles  tangent  to  the  base  line  to  represent 


FIG.    44.— Drawing   a  Cy- 
cloid. 


FIG.  45.— Drawing  an  Epicy- 
cloid. 


successive  positions  of  the  generating  circle  as  it 
is  rolled  along.  With  the  compasses  set  as  for 
spacing  off  the  base  line,  mark  off  one  space  on  the 
arc  which  starts  from  point  1,  two  spaces  on  arc 
2,  three  spaces  on  arc  3,  and  so  forth.  A  curved 

line  starting  at  0  and  pass- 
ing through'  the  points 
thus  obtained  will  be  the 
required  cycloid. 

An  epicycloid,  Fig.  45, 
or  a  hypocycloid,  Fig.  46, 
is  formed  in  precisely  the 
same  way,  excepting  that 
as  the  base  line,  ab,  is  an  arc  of  a  circle,  the  center 
lines  from  points  1-2-3,  etc.,  are  made  radial. 

These  three  cycloidal  curves  are  used  for  the 
shape  of  the  teeth  of  epicycloidal  gears,  sometimes 
called  simply  cycloidal  gears. 


FIG.  46.— Drawing  a  Hypo- 
cycloid. 


GEOMETRICAL  PROBLEMS 


29 


Prob.  22,  Fig.  47.  To  draw  a  parabola  by  means 
of  intersecting  lines. — Draw  the  axis  ax,  and  on  it 
mark  the  focus  /  and  the  vertex  v,  and  at  right 
angles  to  it  draw  the  line  be  at  a  distance  from  v 
equal  to  the  distance  of  v  from/.  Across  the  axis, 
and  at  right  angles  to  it,  draw  a  number  of  lines, 
1,  2,  3,  4,  5,  6.  Then  with  radius  al,  and  with 
center  at  the  focus  /,  draw  arcs  intersecting  line 
1;  with  radius  a2,  and  with  center  again  on /draw 
arcs  intersecting  line  2,  and  so  on.  A  curved  line 


FIG.  47.— Drawing  a  Parabola. 

passing  through  these  intersections  will  be  a  para- 
bola. It  will  be  seen  from  this  method  of  drawing 
a  parabola  that  any  point  on  it  is  equally  distant 
from  the  focus,  and  from  the  line  be,  called  the 
directrix. 

Prob.  23,  Fig.  48.  To  draw  a  parabola  with  a 
pencil  and  string. — Lay  out  the  axis,  the  focus,  the 
vertex  and  the  directrix  as  before.  Attach  one 
end  of  a  thread  to  the  focus,  /  by  means  of  a  pin, 
and  attach  the  other  end  of  the  thread  to  the 
square  shown  at  d,  having  the  thread  of  such 


30 


SELF-TAUGHT  MECHANICAL  DRAWING 


length  that  when  the  inner  edge  of  the  square  is 
on  the  axis,  ax,  the  thread  if  drawn  down  with 
a  pencil  will  just  reach  to  the  vertex,  v.  Now 
slide  the  square  along  be  in  the  direction  of  the 


FIG.  48.— Simplified  Method  of  Drawing  a  Parabola. 

arrow,  keeping  the  pencil  against  the  square;  the 
thread  will  cause  the  pencil  to  move  along  so  as  to 
describe  a  parabola  as  shown. 
Prob.  24,  Fig.  49.    To  draw  a  parabola  of  a  given 


\ 


FIG.  49.— Another  Method  of  Drawing  a  Parabola. 

breadth  of  opening,  ab,  and  of  a  given  depth,  cd.— 
Draw  ef  parallel  with  ab,  and  draw  ae  and  bf  paral- 
lel with  cd,  having  ac  and  be  equal.  Space  off  dc 


GEOMETRICAL  PROBLEMS 


31 


and  df  into  any  number  of  equal  parts,  and  also 
space  off  ea  and/6  into  the  same  number  of  equal 
parts,  as  shown.  From  d  draw  lines  to  the  di- 
visions on  ea  and/6,  and  from  1,  2,  3  and  4  on  de 
and  df  draw  perpendicular  lines  to  intersect  the 
lines  drawn  from  d  to  1,  2,  3  and  4  on  lines  ca  and 
fb.  A  curved  line  passing  through  these  inter- 
sections will  be  the  required  parabola. 

Prob.  25,  Fig.  50.     To  find  the  focus  of  a  para- 
bola.— Let  abed  be  the  given  parabola,  eft  being  its 


FIG.  50.— Finding  the  Focus  of  a  Parabola. 

axis.  Across  the  parabola  at  its  vertex,  v,  draw 
the  line  ij  at  right  angles  to  the  axis.  From  any 
point,  g,  on  the  parabola,  draw  the  line  gh  parallel 
to  the  axis.  With  center  at  g.fmd  a  radius,  by 
trial,  which  will  cut  the  axis  as  much  inside  the 
vertex,  v,  as  it  cuts  the  line  gh  beyond  the  line  ij. 
The  intersection  at  x  will  be  the  required  focus. 


CHAPTER  IV 

PROJECTION 

Mode  of  Representing  Objects. — In  mechanical 
drawing,  machines,  or  parts  of  machines,  are  rep- 
resented by  views,  generally  three,  in  which  per- 
spective is  ignored,  and  which  show  the  object  in 
different  positions  at  right  angles  to  each  other. 
The  mode  of  representing  these  views,  and  their 
positions  with  regard  to  one  another,  which  expe- 
rience has  shown  to  be  most  convenient  is  perhaps 
best  shown  by  means  of  the  familiar  cardboard 
illustration.  Let  abcdefgh,  Fig.  51,  represent  a 
piece  of  cardboard,  which  we  will  suppose  to  be 
transparent,  creased  on  the  dotted  lines  to  permit 
of  the  outer  portions  being  turned  back.  Let  us 
now  suppose  that  we  have  a  prism  shaped  as  shown 
at  C,  and  of  the  length  shown  at  A.  If  the  prism 
is  stood  upright  with  its  broad  side  facing  the  ob- 
server, and  the  cardboard,  being  blank,  is  held  up 
in  front  of  it,  the  prism  will  appear,  if  all  its  lines 
are  brought  perpendicularly  forward  to  the  card- 
board, as  it  is  shown  at  A,  lines  on  the  prism 
which  would  be  hidden  by  its  body,  as  the  further 
corner,  being  dotted.  If  section  Cof  the  cardboard 
is  now  turned  backward  through  an  angle  of  90 
degrees  over  the  top  of  the  prism  we  would  get  the 
view  shown  in  that  part,  all  lines  being  brought 

32 


PROJECTION 


33 


perpendicularly  forward  from  the  prism  to  the 
cardboard  as  before.  Likewise  if  part  D  of  the 
cardboard  were  turned  backward  through  an  angle 
of  90  degrees,  and  the  lines  of  the  prism  were 
brought  perpendicularly  forward  onto  it,  we  would 
get  the  view  shown  in  that  part.  The  view  shown 
at  A  is  called  the  elevation,  that  shown  at  C  is 
called  the  plan,  and  that  shown  at  D  is  called  the 
side  view.  Occasionally  a  piece  is  so  shaped,  or 


FIG.  51. — Principle  of  Projection. 

has  so  much  of  detail  to  it  as  to  make  another  side 
view  desirable;  such  a  view  would  be  placed  at  B. 
In  many  other  cases,  as  in  the  case  of  the  prism 
here  shown,  the  plan  and  elevation  views  alone 
will  fully  show  the  object. 

The  production  of  these  views  from  one  another 
is  called  projection ;  and  by  the  use  of  connecting 
lines,  and  also  at  times  of  temporary  construction 
views,  objects  may  be  shown  at  any  desired  angle, 
irregular  or  curved  lines  may  be  traced,  and  sur- 
faces may  be  developed. 


34 


SELF-TAUGHT  MECHANICAL  DRAWING 


An  Upright  Prism. — Fig.  52  shows  a  prism  in  its 
simplest  position.  A  moment's  examination  will 
show  that  the  elevation  cannot  be  drawn  directly, 
as  the  distance  apart  of  the  vertical  lines  which 
represent  the  corners  of  the  prism,  cannot  be  deter- 
mined without  other  aid;  hence  it  is  necessary  to 
draw  the  plan  view  first.  Horizontal  lines  having 
been  made  to  give  the  height  of  the  prism  in  the 
elevation,  the  vertical  lines  may  then  be  drawn  in 


FIG.   52.— Projections  of 
Prism. 


FIG.  53.— Projections  of 
Tilted  Prism. 


from  the  plan,  as  indicated  by  the  vertical  dotted 
line. 

The  Prism  Inclined  at  One  Angle.— Fig.  53  shows 
the  prism  inclined  to  the  right.  A  brief  exami- 
nation of  these  views  will  show  that  none  of  them 
can  be  drawn  directly,  as  the  distance  apart  of 
the  vertical  lines  in  the  elevation  and  side  views 
is  not  known,  and  the  lines  of  the  plan  view  are 
foreshortened;  but  the  views  can  be  developed 
from  Fig.  52.  It  is  evident  that  as  the  prism  is 
tipped,  the  elevation  view  will  remain  unchanged, 


PROJECTION  35 

hence  the  first  step  will  be  to  reproduce  that  view 
inclined  at  the  desired  angle.  As  the  prism  is 
tipped  it  is  also  evident  that  all  points  in  the  plan 
view  of  Fig.  52  will  move  in  horizontal  lines  to  the 
right,  hence  horizontal  lines  are  drawn  from  these 
points  through  the  position  which  the  plan  will 
occupy  in  Fig.  53.  The  intersection  of  these  lines 
with  vertical  lines  from  the  corresponding  points 
in  the  elevation  will  determine  the  position  of 
each  point  in  the  plan.  The  points  so  determined 
one  by  one  being  then  connected  by  straight  lines, 
gives  the  plan  view  as  shown.  To  make  the  side 
view,  horizontal  lines  are  first  drawn  from  the 
various  points  of  the  prism  as  seen  in  the  eleva- 
tion through  the  position  which  the  side  view  will 
occupy.  Then,  bearing  in  mind  that  each  point 
of  the  prism  in  the  side  view  will  be  as  much  to 
the  left  of  the  vertical  line  ab  as  the  same  point 
in  the  plan  is  below  the  line  ccZ,  the  position  of 
each  point  on  the  horizontal  lines  is  marked  off 
from  ab. 

The  Prism  Inclined  at  Two  Angles. — Fig.  54  shows 
the  prism  tipped  forward  after  having  been  tipped 
to  the  right  as  shown  in  Fig.  53.  An  examination 
of  these  views  will  show  that  not  only  can  they 
not  be  drawn  directly,  but  they  cannot  be  devel- 
oped from  Fig.  52.  They  may,  however,  be  de- 
veloped from  Fig.  53.  It  is  evident  that  as  the 
prism  is  tipped  forward,  the  side  view  of  Fig.  53 
will  remain  unchanged ;  hence  the  first  step  will 
be  to  reproduce  that  view  inclined  at  the  desired 
angle.  Next,  horizontal  lines  are  drawn  from  the 
corners  of  the  prism  as  seen  in  this  view  through 


36 


SELF-TAUGHT  MECHANICAL  DRAWING 


the  place  which  the  elevation  is  to  occupy,  and  the 
perpendicular  line  gh  is  drawn.  It  is  evident  that 
as  the  prism  is  tipped  forward,  the  different  points 
of  it  as  seen  in  the  elevation  of  Fig.  53  do  not 
move  any  to  the  right  or  left,  but  forward  only. 
Hence,  the  distance  of  the  corners  of  the  prism 
from  the  line  ef  may  be  taken  by  the  compasses 
and  marked  off  from  the  line  gh  upon  the  proper 
horizontal  line.  The  new  position  of  all  of  the 

corners  having  thus  been 
determined,  the  con- 
necting straight  lines 
are  drawn,  giving  the 
elevation  as  shown  in 
Fig-,  54.  Vertical  lines 
are  then  drawn  from  the 
different  points  of  the 
prism,  as  seen  in  this 
view,  through  the  posi- 
tion which  the  plan  is 
to  occupy,  and  the  exact 
position  of  each  point 
upon  these  lines  is 

marked  off  from  mn  at  the  same  distance  which 
it  is  from  the  line  jk  in  the  side  view. 

An  Upright  Rectangular  Prism. — The  upright 
rectangular  prism  shown  in  Fig.  55  is,  of  course, 
drawn  in  the  same  way  as  was  the  prism  shown 
in  Fig.  52. 

The  Prism  of  Fig.  55  Tipped  Forward  on  One  Edge. 
— It  is  evident  that  if  the  prism  were  to  be  tipped 
on  its  edge  in  the  direction  of  the  arrow  No.  1,  the 
result  would  be  the  same  as  though  it  had  been 


FIG.  54.— Projections  of  Prism 
Tilted  in  Two  Directions. 


PROJECTION 


tipped  first  to  the  right,  and  then  directly  forward, 
as  was  done  to  produce  Fig.  54;  but  as  those 
angles  are  not  given,  the  method  employed  in  that 
case  is  not  readily  available. 

Fig.  56  shows  the  prism  tipped  to  its  new  po- 
sition, and  shows,  also,  the  method  employed  to 
produce  the  views.  Draw  the  line  cd  at  the  same 


FIG.  55.— Upright   Rectan- 
gular Prism. 


FIG.  56.— Rectangular  Prism 
Tipped  Forward. 


angle  to  the  horizontal  as  the  edge  ab  of  the  prism 
in  Fig.  55,  and  make  e/at  right  angles  to  it.  Upon 
these  lines  draw  the  temporary  side  view  of  the 
prism,  A,  tipped  at  the  desired  angle.  With  the 
aid  of  this  view  the  plan  view  is  readily  drawn. 
Vertical  lines  are  then  drawn  from  the  various 
points  of  the  plan  view  through  the  place  which 
the  elevation  is  to  occupy,  and  the  exact  location 
of  each  point  is  marked  off  on  these  lines  at  the 


38 


SELF-TAUGHT  MECHANICAL  DRAWING 


same  height  above  the  base  line  gh  that  it  is  above 
the  line  ef  in  the  temporary  side  view,  A.  The 
permanent  side  view  is  then  developed  from 
the  plan  and  elevation  in  the  same  way  as  was  the 
side  view  of  Fig.  53. 

Let  it  now  be  required  to  tip  the  prism  of  Fig.  55 
forward  on  one  corner  in  the  direction  of  arrow 
No.  2. 

It  will  be  seen  that  tipping  it  in  this  direction 


FIG.  57. — Rectangular  Prism  Tipped  in  Two  Directions. 

will  cause  a  foreshortening  of  all  of  the  lines  in  the 
plan,  hence  the  use  of  a  single  temporary  view 
such  as  was  used  in  Fig.  56  will  not  solve  the 
problem;  but  it  may  be  solved  by  the  use  of  two 
temporary  views  as  shown  in  Fig.  57.  Draw  the 
line  ab  in  the  direction  in  which  the  prism  is  to  be 
tipped,  and  the  line  cd  at  right  angles  to  it.  At  A 
reproduce  the  plan  view  of  Fig.  55,  and  at  B  draw 


PROJECTION 


39 


a  side  view  of  the  prism  as  it  would  appear  if  A 
were  viewed  in  the  direction  of  the  arrow,  but  in- 
clined to  cd  at  the  required  angle.  The  intersec- 
tion of  lines  drawn  from  the  corners  of  A,  parallel 
with  ab,  with  lines  drawn  from  the  same  corners 
of  B,  parallel  with  cd,  will  give  their  location 


FIG.  58.— Projections  of  a  Cube. 

in  the  permanent  plan  view.  This  view  being 
finished,  the  elevation  and  the  permanent  side 
views  are  drawn  in  the  same  way  as  were  those 
of  Fig.  56. 

Let  a  cube  be  set  on  one  corner  so  that  a  diagonal 
of  it  shall  be  horizontal;  required  to  show  the  angle 
which  the  edges  that  meet  at  that  forward  corner 
make  with  a  plane  perpendicular  to  the  diagonal, 
the  angle  which  the  sides  that  have  corners  coming 
together  at  the  same  point  make  with  the  plane,  and 


40 


SELF-TAUGHT  MECHANICAL  DRAWING 


also  the  am  ^  ant  of  foreshortening  of  the  lines  which 
will  be  caused. 

In  Fig.  58,  A  shows  a  face  view  of  the  cube  set 
on  edge,  B  shows  a  side  view  of  the  same,  and  C 
shows  B  inclined  until  the  diagonal  ke  becomes 
horizontal.  The  length  of  ke  being  laid  out  on  the 
center  line,  the  position  of  the  other  corners  is  ob- 
tained as  indicated  by  the  arcs  a,  b,  c  and  d.  The 
angle  geh  is  the  required  angle  which  the  edges 
•  which  meet  at  e  make 

with  a  plane  perpendic- 
ular to  ek,  of  which  fg  is 
an  edge  view ;  the  angle 
fej  is  the  angle  which 
the  sides  having  corners 
meeting  at  e  make  with 
the  plane.  D  is  a  face 
view  of  C,  and  any  of  its 
lines,  when  compared 
with  any  of  the  lines  of 
A,  will  show  the  fore- 
shortening caused  by  the 
cube  being  put  into  this 
position. 

The  Surface  Develop- 
ment of  a  Cone. — Let  A 
and  B,  Fig.  59,  be  the 
plan  and  elevation  views 
of  a  cone.  With  a  radius 
equal  to  ab,  and  with  a 

center  at  c,  draw  the  arc  def,  making  it  equal  in 
length  to  the  circumference  of  the  base  of  the  cone, 
as  shown  at  A,  This  may  be  most  conveniently 


FIG.  59.— Development  of 
a  Cone. 


PROJECTION 


41 


done  by  spacing  it  off.  Draw  the  lines  cd  and  cf, 
and  the  figure  C  thus  formed  will  be  the  required 
surface  development. 

The  Surface  Development  of  a  Pyramid  Having 
Its  Top  Cut  Off  Obliquely.— In  Fig.  60,  A,  B  and  C 
show,  respectively,  the  plan,  elevation,  and  side 


FIG.  60.— -Development  of  a  Frustum  of  a  Pyramid. 

views  of  the  pyramid,  the  top  of  which  is  cut  off  by 
the  plane  ab.  These  views  may  be  made  by  the 
principles  already  explained,  as  may  also  the  view 
at  D,  which  shows  the  pyramid  as  though  B  were 
viewed  in  the  direction  of  the  connecting  dotted 
line,  which  is  at  right  angles  to  ab,  thus  showing 
the  shape  of  the  section  exposed  by  cutting  off  the 
top. 


42  SELF-TAUGHT  MECHANICAL  DRAWING 

To  get  the  surface  development,  take  a  radius 
equal  to  the  length  of  one  edge  of  the  pyramid  as 
shown  at  cd  in  the  elevation,  this  being  the  only 
one  which  shows  at  full  length,  the  others  being 
more  or  less  foreshortened,  and  with  a  center  at  e 
in  view  E,  draw  an  arc  of  a  circle  upon  which  the 
sides  of  the  base  are  to  be  marked  off.  These 
points  are  connected  with  one  another  and  with  e; 
this  gives  the  shape  of  the  surface  of  the  whole 
pyramid.  Upon  the  lines  connecting  the  points 
with  e,  as  el,  e2  and  e3,  the  .lengths  of  the  different 
edges  of  the  cut  off  pyramid  are  marked  off.  As 
the  edge  which  is  seen  at  the  left  in  the  elevation 
shows  full  length,  its  length,  dl,  may  be  taken  di- 
rectly and  marked  off  on  the  line  el.  As  the  other 
edges  are  seen  foreshortened,  their  lengths  cannot 
be  taken  directly,  but  by  horizontally  transferring 
the  upper  end  of  each  edge  to  the  line  cd,  their 
actual  lengths  d2  and  d3  may  be  obtained  and  then 
marked  off  on  the  lines  e2  and  e3.  The  points  so 
obtained  being  connected,  and  the  outer  half  sec- 
tions being  finished,  gives  the  required  surface 
development. 

If  the  cone  shown  in  Fig.  59  were  to  have  its  top 
cut  off  obliquely,  the  views  of  it  corresponding  to 
A,  B,  Cand  D,  Fig.  60,  and  its  surface  develop- 
ment, would  be  obtained  by  dividing  off  its  base, 
as  seen  in  the  plan,  into  any  number  of  sides,  and 
then  proceeding  as  though  it  were  a  pyramid  of 
that  number  of  sides,  until  the  points  correspond- 
ing to  those  of  Fig.  60  had  been  located,  but  then 
connecting  them  with  curved  lines  instead  of 
straight  lines. 


PROJECTION 


43 


Intersecting  Cylinders,  Fig.  61.— Required  the 
line  of  the  intersection,  the  surface  development  of  the 
branch,  and  the  shape  which  the  end  of  the  branch 
would  appear  to  have  as  seen  in  the  view  at  the 
right. 

First  draw  the  elevation,  A,  in  outline,  and  as 
much  of  the  end  view,  B,  as  can  be  directly  drawn. 


FIG.  61. — Intersecting  Cylinders. 

Opposite  the  end  of  the  branch  in  each  of  these 
views,  and  in  line  with  it,  draw  circles  of  the  same 
diameter  as  the  branch,  and  space  off  the  semi- 
circumference  nearest  to  it  into  a  number  of 
equal  parts,  the  same  number  in  both  cases.  From 
the  points  so  obtained  draw  lines  parallel  with  the 
center  line  of  the  branch,  as  shown.  From  the 
points  where  these  lines  in  the  view  B  meet  the 


44  SELF-TAUGHT  MECHANICAL  DRAWING 

circle  representing  the  end  of  the  large  cylinder, 
draw  horizontal  lines  intersecting  the  lines  drawn 
from  C.  These  intersections  will  be  points  through 
which  the  line  of  the  intersection  of  the  cylinders 
is  to  be  drawn.  From  the  points  where  the  lines 
drawn  from  C  cross  the  end  of  the  branch,  draw 
horizontal  lines  intersecting  those  drawn  from  D. 
These  intersections  will  be  points  through  which 
the  line  representing  the  end  of  the  branch  is  to 
be  drawn. 

To  get  the  surface  development  of  the  branch, 
first  draw  the  line  ab,  in  E,  having  it  in  line  with 
the  end  of  the  branch.  Make  this  line  equal  in 
length  to  the  circumference  of  the  branch,  spacing 
it  off  equally  each  way  from  the  center  line  OX  into 
the  same  number  of  spaces  as  the  semi -circumfer- 
ence of  C  was  divided  into.  From  these  points 
draw  lines  parallel  with  OX,  and  from  the  points 
in  the  intersection  of  the  two  cylinders,  previously 
obtained,  draw  lines  parallel  with  ab,  intersecting 
these  lines.  These  intersections  will  be  points 
through  which  a  curved  line  is  to  be  drawn,  thus 
giving  the  completed  surface  development  of  the 
branch. 

In  drawing  these  curved  lines  through  the  points 
of  intersection,  the  irregular  curves  mentioned  in 
the  early  part  of  the  chapter  on  instruments  and 
materials  are  used. 

Intersecting  Cylinder  and  Frustum  of  Cone,  Fig. 
62. — Required  line  of  intersection  and  surface  de- 
velopment of  branch,  as  before. 

Draw  the  elevation,  A,  in  outline,  continuing  the 
sides  of  the  conical  branch  either  way  until  they 


PROJECTION 


45 


meet  at  their  vertex,  a,  on  the  one  hand,  and  to 
any  convenient  points,  c  and  d,  on  the  other.  In  a 
similar  manner  draw  as  much  of  the  end  view,  B, 


FIG.  62.— Intersecting  Cylinder  and  Cone. 

as  can  be  made  directly.  With  centers  at  a  and  at 
6,  and  with  any  convenient  radius,  draw  the  arcs 
c'd  and  ef,  intersecting  the  extended  sides  of  the 
conical  branch.  Then,  with  centers  at  the  inter- 


46  SELF-TAUGHT  MECHANICAL  DRAWING 

section  of  these  arcs  with  the  center  line  of  the 
branch,  draw  the  half  ci cries  shown,  tangent  to 
the  extended  sides  of  the  branch,  and  space  them 
off  into  a  number  of.  equal  parts,  the  same  number 
in  each  case.  From  these  points  draw  lines  to  the 
vertices  a  and  b.  From  the  points  where  these 
lines  in  the  end  view,  B,  intersect  the  circle  repre- 
senting the  end  of  cylinder,  draw  horizontal  lines 
to  the  elevation,  A,  intersecting  the  lines  drawn 
from  the  vertex  a  to  the  half -circle  cd.  The  inter- 
sections will  be  points  through  which  the  line  rep- 
resenting the  intersection  of-  the  cylinder  and  its 
conical  branch  is  to  be  drawn.  The  shape  of  the 
end  of  the  branch  as  seen  in  the  end  view,  B,  is 
now  obtained  in  the  same  manner  as  in  the  case  of 
the  intersecting  cylinders.  From  the  points  where 
the  lines  drawn  from  the  vertex,  a,  of  the  side  ele- 
vation A,  to  the  half-circle  at  cd,  cross  the  end  of 
the  branch,  draw  horizontal  lines  intersecting  the 
lines  drawn  from  the  vertex  b.  These  intersec- 
tions will  give  points  through  which  the  line  rep- 
resenting the  end  of  the  branch  in  view  B  is  to 
be  drawn. 

To  get  the  development  of  the  branch  as  shown 
at  F  take  a  radius  equal  to  the  distance  from  the 
apex  a  to  the  end  of  the  branch  as  seen  in  the  side 
elevation,  A,  and  with  a  center  at  g  draw  an  arc 
hi,  making  the  length  of  the  arc  equal  to  the  cir- 
cumference of  the  end  of  the  branch  as  shown  at 
E,  spacing  equally  each  way  from  the  center  line 
gj,  the  length  and  number  of  the  spaces  each  way 
being  the  same  as  those  obtained  in  spacing  off  the 
semicircle  at  E.  Through  these  points  draw  lines 


PROJECTION 


47 


radiating  from  g,  as  shown.  On  these  lines  dis- 
tances are  marked  off  from  the  arc  hi  through 
which  the  irregular  curved  line  is  drawn  which 
gives  the  development  of  the  branch.  The  lengths 
at  the  middle  and  at  the  extremities  may,  of  course, 
be  taken  directly  from  the  elevation  A,  the  length 
kl  being  the  length  on  the  center  line,  and  the 
length  mn  being  the  length  at  the  extremities. 
The  other  lengths,  being  foreshortened,  as  seen  in 
the  elevation  A,  cannot  be  taken  directly,  but  are 
obtained  by  transferring  the  points  to  either  kl  or 
mn  as  shown  by  the  dotted  lines,  as  was  done  in 
the  case  of  the  pyramid,  Fig.  60. 

To  Draw  a  Helix. — A  helix  is  a  line  of  such  shape 
as  would  be  made  by  winding  a  thread  around  a 


FIG.  63.— Drawing  a  Helix. 


FIG.  64.— The  Helix  as 
it  Appears  in  a  Screw 
Thread. 


cylinder,  and  having  it  advance  lengthwise  on  the 
cylinder  at  a  uniform  rate  as  it  is  wound  around 
it.  In  Fig.  63  we  have  the  side  and  end  views  of 
a  cylinder  upon  which  it  is  desired  to  draw  a  helix, 
which  shall  advance  from  a  to  b  in  making  a  half 
turn  around  it.  Divide  the  space  from  a  to  b  into 
any  number  of  equal  parts,  and  at  the  points  so 
obtained  erect  perpendicular  lines.  Divide  the 


48  SELF-TAUGHT  MECHANICAL  DRAWING 

semi-circumference  of  the  end  view  of  the  cylinder, 
toward  the  side  view,  into  the  same  number  of 
equal  parts,  and  from  these  points  draw  horizontal 
lines  to  meet  the  perpendiculars  previously  erected. 
Where  these  lines  meet  will  be  points  through 
which  the  helix  is  to  be  drawn. 

The  outlines  of  a  screw  thread  are  helices.  Fig. 
64  shows  a  double  threaded  Acme  standard,  or  29 
degree  threaded  screw,  the  outline  of  which,  on 
its  outside  diameter,  is  the  helix  of  Fig.  63. 

Isometric  Projection. — If  a  cube  is  tipped  over  on 
one  corner,  so  that  the  diagonal  of  it  is  horizontal 
as  shown  at  D,  Fig.  58,  and  also  in  Fig.  65,  the 


FIG.  65.— Principle  of  FIG.  66.— An  Example  of 

Isometric  Projection.  Isometric  Projection. 

lines  of  it  will  all  appear  of  equal  length.  Draw- 
ings made  on  this  principle,  as  Fig.  66,  are  called 
isometric  drawings.  Vertical  lines  remain  ver- 
tical. Horizontal  lines  become  inclined  to  the 
horizontal  of  the  paper  at  an  angle  of  30  degrees. 
Circles  appear  as  ellipses,  which  may  be  drawn  as 
shown  in  the  upper  square  of  Fig.  65.  From  the 
ends  of  the  "short"  diagonals,  lines  are  drawn  to 
the  middle  of  the  opposite  sides.  Where  these 
lines  cross  the  "long"  diagonals  are  located  the 
centers  from  which  the  ends  of  the  ellipse  may 


PROJECTION  49 

be  drawn.  The  ends  of  the  short  diagonals  will 
be  centers  from  which  to  draw  the  sides  of  the 
ellipse. 

Irregular  curves  may  be  drawn  as  indicated  in 
Figs.  67  and  68.  The  figure  2  there  shown  is  first 
drawn  in  the  desired  position  in  a  naturally  shaped 
square,  which  is  then  divided  off  by  equally  spaced 
lines  into  smaller  squares.  The  isometric  square 
is  then  similarly  divided  off,  and  the  figure  is 


FIGS.  67  and  68.— Method  of  Transferring  Irregular  Lines  in 
Isometric  Projection. 

made  to  pass  through  the   corresponding  inter- 
sections. 

Isometric  drawings  differ  from  perspective  draw- 
ings in  that  receding  lines  remain  parallel,  instead 
of  converging  to  a  vanishing  point.  They  may  be 
measured  the  same  as  ordinary  drawings  in  any 
one  of  the  three  directions  indicated  by  the  lines 
of  the  cube.  The  foreshortening  of  the  lines  caused 
by  tipping  the  cube  into  this  position  is  generally 
ignored.  If  an  isometric  drawing  is  to  be  shown 
in  connection  with  ordinary  views,  however,  it 
should  be  made  on  a  scale  of  about  8-10  of  an  inch 
to  the  inch,  otherwise  it  would  appear  too  large. 


CHAPTER  V 

WORKING    DRAWINGS 

As  the  object  of  working  drawings  is  to  convey 
to  the  workman  a  clear  idea  of  the  appearance  and 
construction  of  the  piece  to  be  made,  and  as  the 
whole  "science"  of  mechanical  drawing  has  been 
developed  primarily  for  the  purpose  of  conveying 
the  ideas  and  thoughts  of  the  designer  and  drafts- 
man to  the  men  who  carry  out  these  ideas  in  wood 
and  metal,  the  subject  of  working  drawings  is  of 
supreme  importance  to  all  mechanics.  A  working 
drawing  should  be  as  complete  as  possible,  so  com- 
plete, in  fact,  that  when  it  has  once  passed  out  of 
the  draftsman's-  hand  into  the  shop,  no  further 
questions  will  be  necessary.  In  or.der  to  accom- 
plish this,  all  necessary  information,  of  whatever 
kind,  should  be  .included,  and,  if  required,  short 
notes  and  directions  may  be  written  on  the  draw- 
ing to  prevent  eventual  misunderstandings. 

The  number  of  views  necessary  to  properly  rep- 
resent an  object  must  be  left  for  the  draftsman's 
judgment  to  determine.  Usually  two  views  are 
sufficient,  when  the  object  is  simple,  but  when  at 
all  complicated,  three  or  more  views  will  be  found 
necessary.  Cylindrical  pieces  can  often  be  ade- 
quately represented  by  a  single  view,  on  which  the 
various  diametral  and  length  dimensions  are  given. 

50 


WORKING  DRAWINGS  51 

While  it  is  customary  to  put  the  plan  view  of  an 
object  above  the  elevation,  it  frequently  becomes 
necessary,  in  order  to  present  the  objects  shown  in 
as  clear  a  manner  as  possible,  to  deviate  from  this 
rule.  A  case  of  this  kind  is  shown  in  Fig.  69, 
where  the  shaft  hanger  illustrated  has  been  se- 
lected as  an  example  of  the  methods  employed  in 
working  drawings. 

An  examination  of  the  hanger  will  show  that  if 
the  plan  were  placed  above  the  elevation,  and  if  it 
were  represented  according  to  the  methods  already 
explained,  the  box  and  the  yoke  with  its  adjusting 
screws  and  check-nuts  would  have  to  be  shown 
mostly  by  dotted  lines.  Such  a  multiplicity  of 
dotted  lines  would  tend  to  confusion;  hence  the 
object  in  view,  that  of  presenting  the  hanger  in  as 
clear  a  manner  as  possible,  is  best  accomplished  in 
a  case  like  this  by  having  the  plan  underneath  the 
elevation,  and  letting  it  be  a  bottom  view  instead 
of  a  top  view. 

In  designing  a  machine  detail  of  this  kind,  the 
starting  point  would  of  necessity  be  the  shaft 
itself,  and  the  first  step  would  be  to  design  the 
box;  next  would  come  the  yoke,  and  lastly,  the 
frame.  Much  of  the  preliminary  work  may  fre- 
quently be  done  on  scrap  paper;  having  determined 
the  size  and  proper  proportions  of  the  various 
parts,  the  position  which  the  different  views  will 
occupy  in  the  finished  drawing  is  easily  ascer- 
tained. The  center  lines  are  then  laid  out  as 
shown,  and  the  drawing  built  up  about  these  lines 
as  a  base. 

When  a  drawing  is  for  temporary  use  only,  and 


52 


SELF-TAUGHT  MECHANICAL  DRAWING 


the  mechanism  represented  on  it  of  a  simple  nature, 
the  assembly  drawing,  corresponding  to  the  three 
views  in  Fig.  69,  will  answer  all  purposes,  the  di- 
mensions being  given  directly  on  this  drawing.  In 


FIG.  69.— Shaft  Hanger. 

some  cases  only  the  most  important  dimensions 
would  be  given,  those  of  secondary  consequence 
being  left  for  the  workman  to  be  obtained  by 
"  scaling' '  the  drawing.  This  procedure,  however, 
is  possible  only  when  the  drawing  is  made  care- 
fully to  scale,  and  is  not  one  that  should  be  en- 


WORKING  DRAWINGS 


53 


couraged.  In  general,  a  drawing  should  be  so  di- 
mensioned that  it  can  be  worked  to  without  the 
workman  obtaining  any  measurements  by  "scal- 
ing" the  drawing. 

In  most  cases  it  is  not  possible  to  show  the  de- 
tails of  a  mechanism  clearly  enough  in  an  assembly 
drawing;  for  if  the  device  shown  is  more  or  less 
complicated,  a  hopeless  confusion  results  from  the 
attempt  to  put  in  all  the  lines  necessary  to  fully 
show  all  the  details ;  neither  would  it  be  possible, 
for  the  same  reason,  to  give  more  than  the  princi- 


CAST  IRON,  BABBITTED 

FIG.  70.— Example  of  Working  Drawing. 

pal  dimensions.  In  such  cases  it  is,  therefore,  cus- 
tomary, after  the  assembly  drawing  has  been  com- 
pleted, and  the  proper  sizes  and  proportions  of  the 
various  parts  of  the  mechanism  thus  ascertained, 
to  make  a  separate  drawing  of  each  detail,  either 
on  the  same  sheet  of  paper,  or  on  separate  sheets. 
This  permits  the  parts  of  the  mechanism  to  be 
clearly  and  completely  shown  and  fully  dimen- 
sioned. Figs.  70  and  71  show  two  pieces  of  the 
hanger  in  Fig.  69  detailed  in  this  manner.  These 
detail  drawings  give  all  the  required  informa- 
tion for  the  making  of  the  pieces,  and  the  assembly 


54 


SELF-TAUGHT  MECHANICAL  DRAWING 


drawing  merely  shows,  in  a  general  way,  how  the 
parts  are  to  be  assembled  when  completed. 

In  the  case  of  jig  and  fixture  drawings,  it  is  the 
practice  in  a  great  many  large  drafting-rooms  to 
show  assembled  views  only,  and  to  put  all  dimen- 


CAST  IRON, 


Tap  %"- 
10  thd. 


FIG.  71.— Example  of  Working  Drawing. 

sions  directly  on  the  assembly  drawing;  the  argu- 
ment advanced  in  favor  of  this  practice  is  that  ex- 
perienced pattern-makers  and  tool-makers,  who  are, 
as  a  rule,  the  only  mechanics  who  will  work  on 
the  making  of  these  tools,  will  find  no  difficulty  in 
reading  the  assembly  drawing;  besides,  it  is  said, 


WORKING  DRAWINGS  55 

as  a  drawing  of  this  kind  is,  in  most  cases,  used 
but  once,  it  would  be  waste  of  time  to  have  the 
draftsman  detail  the  different  parts  of  the  tool. 

While  these  arguments  are  undoubtedly  true  in 
the  case  of  very  simple  jigs  and  fixtures,  there  can 
be  little  doubt  that  in  the  case  of  more  complicated 
ones,  the  comparatively  short  time  required  by  the 
draftsman  to  make  detail  drawings  will  be  saved 
many  times  over  in  the  shop;  for  the  pattern- 
maker and  tool-maker  will  not  have  to  spend,  in 
the  total,  a  number  of  hours  puzzling  over  the  draw- 
ing, and  even  then  being  liable  to  make  a  mistake. 

In  making  drawings,  it  is  always  a  rule  to  work 
from  the  center  lines,  when  the  outline  of  the 
piece  is  such  that  it  has  a  definite  center  line. 
Dimensions  in  either  direction  from  the  center 
line  can  be  best  marked  off  with  the  compasses. 
This  insures  a  symmetrical  appearance  to  the  fin- 
ished drawing,  such  as  might  not  be  secured  if  the 
dimensions  are  set  off  on  either  side  of  the  center 
line  from  the  rule,  it  always  being  easy  to  then 
introduce  small  errors  which  show  plainly  in  the 
finished  work.  If  the  piece  is  of  such  shape  as  to 
have  no  center  line,  some  one  principal  line  may 
be  selected,  one  in  each  direction  in  each  view, 
and  the  remaining  points  and  lines  may  be  located 
from  these  lines. 

The  various  styles  of  lines  ordinarily  used  in 
working  drawings  are  shown  in  Fig.  72.  The 
regular  "full"  line  A  A  is  used  for  the  outlines  of 
objects,  and  when  drawn  rather  "fine,"  for  cross- 
hatching  or  cross-sectioning.  The  heavy  shade 
line  BB  is  used  to  represent  lines  assumed  to  sepa- 


56  SELF-TAUGHT  MECHANICAL  DRAWING 

rate  the  light  surfaces  of  an  object  from  the  dark, 
as  will  be  explained  in  the  following.  The  dotted 
line  CC,  as  has  already  been  explained  in  the  pre- 
vious chapter,  is  used  to  represent  lines  obscured 
or  hidden  from  view.  The  line  DD,  called  a 
"dash"  line,  is  used  by  a  great  many  draftsmen 
for  dimension  lines.  Finally,  the  line  EE,  the 
"dash  and  dot,"  or,  simply,  the  "dash-dotted" 


FIG.  72.— Styles  of  Lines  Used  on  Working  Drawings. 

line,  is  used  in  common  practice  for  center  lines, 
to  indicate  sections,  etc.  This  line  is  also  com- 
monly used  for  construction  lines,  in  laying  out 
mechanical  movements. 

The  dimension  lines  may  be  made  either  fine 
full  lines  or  "dash"  lines,  the  dashes  being  about 
|  inch  long.  A  space  is  left  open  for  the  figures 
giving  the  dimension.  The  witness  points  or  ar- 
row heads,  showing  the  termination  of  the  dimen- 
sion, are  made  free  hand.  Many  draftsmen  draw 
the  extension  and  dimension  lines  in  red  ink,  the 
arrow  heads,  however,  still  being  made  black.  It 
is  well  to  avoid,  as  far  as  possible,  having  the 


WORKING  DRAWINGS  57 

dimension  lines  cross  each  other,  as  such  crossing 
tends  to  confusion;  the  difficulty  can  usually  be 
avoided  by  having  at  least  one  set  of  dimensions 
placed  outside  or  between  the  views,  the  larger  di- 
mensions being  placed  farther  from  the  outline  of 
the  object  than  the  shorter  ones,  to  avoid  having 
the  extension  lines  of  the  latter  cross  the  dimen- 
sion lines  of  the  former.  Dimensions  under  24 
inches  are  most  conveniently  given  in  inches; 
larger  dimensions  are  given  in  feet  and  inches. 
The  usual  practice  is  to  indicate  feet  and  inches 
on  drawings  by  short  marks,  "  prime' '  marks  ('), 
placed  at  the  right,  and  a  little  above  the  figure, 
one  mark  (0  indicating  feet,  and  two  marks, 
" double  prime' '  marks  ("),  indicating  inches,  so 
that  5'  7"  would  read  5  feet  1  inches.  Some  drafts- 
men do  not  consider  this  method  of  marking  safe 
enough  to  eliminate  mistakes,  and  prefer  to  write 
dimensions  of  this  kind  in  the  form  5  ft.  7".  A 
method  equally  satisfactory  in  preventing  possible 
mistakes  is  to  place  a  short  dash  between  the 
figure  giving  the  number  of  feet  and  that  giving 
the  number  of  inches,  at  the  same  time  retaining 
the  "prime"  marks;  thus,  5'— 7".  When  feet  only 
are  given,  it  is  well,  for  the  sake  of  uniformity 
and  to  prevent  any  misunderstanding,  to  give  the 
dimension  in  the  form  5' — 0". 

A  few  examples  showing  the  principles  of  the 
usual  methods  of  dimensioning  drawings  may  be 
of  value.  In  Fig.  73  is  shown  a  simple  bushing. 
The  diameter  of  the  hole  or  bore  is  given  as  2 
inches  by  a  dimension  line  passing  through  the 
center  of  the  circles  in  the  end  view.  It  is  con- 


58 


SELF-TAUGHT  MECHANICAL  DRAWING 


fusing,  however,  to  have  more  than  one  dimension 
line  passing  through  the  same  center,  and,  there- 
fore, the  outside  diameters  of  the  bushing  have 
been  given  on  the  side  view.  The  lengths  of  the 
various  steps  or  shoulders  of  the  bushing  are  given 
below  the  side  view,  as  is  also  the  total  length.  It 
will  be  noticed  that  the  dimensions  of  the  three 
steps  are  slightly  offset — that  is,  the  dimension 


FIG.  73. — Simple  Example  of  Dimensioning  a  Drawing. 

lines  do  not  extend  in  one  straight  li'ne ;  this  makes 
a  very  clear  arrangement. 

The  method  of  dimensioning  holes  drilled  in  a 
circle  is  shown  in  Fig.  74.  Outside  of  the  dimen- 
sion for  the  holes  themselves  only  the  diameter  of 
the  circle  passing  through  the  centers  of  the  holes 
is  given,  together  with  the  number  of  holes.  As 
the  holes,  of  course,  are  to  be  equally  spaced,  that 
is*  all  that  is  required.  When  a  great  many  bolt 
holes  or  bolts  occur  around  a  flange,  it  is  not  nec- 
essary to  draw  them  all  in  on  the  working  draw- 
ing; a  common  method  is  to  show  a  few,  and  to 


WORKING  DRAWINGS 


59 


draw  the  circle  passing  through  their  centers,  the 
pitch  circle.  The  total  number  of  bolts  around  the 
flange  is,  of  course,  also  given.  A  case  of  this 
kind  is  illustrated  in  Fig.  75.  When  a  great  many 
holes  are  drilled  in  a  row,  a  similar  expedient  may 


FIG.  74.— Dimensioning  Holes  Drilled  in  a  Circle. 

be  adopted  to  avoid  showing  and  dimensioning 
all  the  holes;  an  illustration  of  this  is  shown  in 
Fig.  76. 

In  Fig.  77  are  shown  the  common  methods  of 
dimensioning  screws  and  bolts.  At  A  is  shown  a 
hexagon  head  bolt,  so  drawn  that  three  sides  of 


60 


SELF-TAUGHT  MECHANICAL  DRAWING 


the  head  are  visible.  Hexagon  bolt-heads  are 
usually  drawn  in  this  manner  in  all  views,  irre- 
spective of  the  fact  that  the  rules  of  projection 
would  call  for  only  two  sides  to  be  visible  in  one 
view.  The  reason  for  this  is  partly  that  the  bolt- 


FiG.  75. — Simplified  Method  of  Dimensioning  Holes 
Drilled  in  a  Circle. 

head  looks  better  when  three  sides  are  visible,  and 
partly  that  when  so  drawn  there  can  be  no  confu- 
sion whether  a  hexagon  or  a  square  head  is  meant. 
If  only  two  sides  were  shown,  as  at  B,  the  head, 
especially  if  carelessly  drawn,  might  be  mistaken 
for  a  square  bolt-head.  As  a  rule,  the  dimensions 


WORKING  DRAWINGS  61 

of  bolt-heads  are  standard  for  given  diameters  of 
bolts,  and  no  dimensions  are  required  for  the 
head.  In  some  cases,  however,  the  head  may  be 
required  to  fit  a  given  size  of  wrench,  or  for  some 
other  reason  be  required  to  be  made  different  from 
the  standard  size ;  in  such  cases  dimensions  may 
be  given  as  shown  at  C,  Fig.  77,  the  dimension 
"V  hex."  indicating  that  the  head  is  one  inch 


K 10-HOLE8-2-CENTER-DISTANCE »j 


! 

* 

~t- 

--$--&- -6 

j< : 10-HOLES-2-CENTER-DISTANOE >j 


T 


FIG.  76. — Dimensioning  Holes  Drilled  in  a  Row. 

"across  flats."  In  the  same  way,  "f"  sq."  would 
indicate  that  the  head  should  be  square,  and  three- 
quarters  inch  "across  flats." 

The  length  of  the  bolt  should  be  given  as  shown 
in  the  lower  view  in  Fig.  77.  The  dimensions 
should  be  given  "under  the  head,"  both  the  total 
dimension,  and  the  distance  to  the  beginning  of 
the  thread. 

In  general,  full  circles  should  be  dimensioned  by 
their  diameters ;  an  arc  of  a  circle,  again,  should 
be  dimensioned  by  its  radius.  The  center  from 


62 


SELF-TAUGHT  MECHANICAL  DRAWING 


which  the  arc  is  struck  should  preferably  be  indi- 
cated by  a  small  circle  drawn  around  it.  In  small 
dimensions,  the  arrow  points  are  frequently  placed 
outside  of  the  lines  between  which  the  dimension 
is  given,  as  shown  in  Fig.  71  in  dimensioning 
the  narrow  ribs;  sometimes,  the  figures  giving  the 


"A"7 

^ 

JL 


-a 


8T  D 
HEAD 


<  --        ---  2M— 


FIG.  77. — Dimensioning  Screws  and  Bolts. 

dimension  are  themselves  placed  outside  of  the 
space  between  the  arrow  heads,  because  the  space 
is  too  small  to  permit  the  dimension  to  be  clearly 
written  within  it. 

The  principal  dimensions  should  be  so  given 
that  the  workman  will  not  have  to  add  a  number 
of  other  dimensions  to  get  them.  When  the 
dimensioning  of  a  piece  naturally  divides  itself 
into  several  measurements,  an  over-all  dimension 
should  always  be  given  for  verification.  If,  how- 


WORKING  DRAWINGS  63 

ever,  the  piece  terminates  with  a  round  end,  as 
the  yoke  in  Fig.  71,  the  over-all  dimension  may 
properly  terminate  at  the  center  of  curvature  of 
the  end,  the  distance  beyond  being  of  entirely 
secondary  importance,  and  being  taken  care  of  by 
its  radius.  If  a  dimension  has  been  given  in  one 
view,  there  is  usually  no  reason  for  repeating  it 
in  the  other  views;  sometimes  such  repetitions 
would  cause  too  many  dimensions  to  be  given  in 
each  view,  so  that  confusion  would  arise,  and  in- 
stead of  making  the  drawing  plainer,  the  repeti- 
tion of  dimensions  might  cause  mistakes  which 
otherwise  would  have  been  avoided. 

Drawings  should  always  be  dimensioned  the  full 
size  of  the  finished  article,  regardless  of  the  scale 
to  which  the  drawing  is  made.  If  a  drawing  is 
made  to  any  other  scale  than  full  size,  it  is  cus- 
tomary to  state  on  the  drawing  the  scale  to  which 
it  is  made,  as  " Scale,  i  inch=l  ft." 

A  drawing  should  be  so  marked  as  to  tell  the 
workman  what  surfaces  are  to  be  finished ;  a  fin- 
ished surface  is  usually  indicated  by  the  letter 
"f"  placed  either  upon  the  line  representing  the 
surface,  or  in  close  proximity  to  it.  While  the 
amount  and  kind  of  finish  is  usually  left  to  the 
workman  to  determine,  the  best  modern  methods 
require  that  the  draftsman  should  indicate  on  the 
drawing  how  closely  the  various  parts  are  to  be 
machined.  A  very  commendable  method  is  to 
give  dimensions  in  thousandths  of  an  inch,  where 
accuracy  is  required,  and  in  common  fractions  in 
cases  where  there  is  no  need  of  working  to  thou- 
sandths. In  very  highly  systematized  establish- 


64  SELF-TAUGHT  MECHANICAL  DRAWING 

ments,  the  limits  of  variation  between  which  any 
measurement  is  allowed  to  vary,  are  given  with 
each  dimension,  or,  at  least,  with  dimensions  for 
diameters  which  are  to  fit  the  holes  or  bores  of 
other  pieces.  The  determination  of  the  limits  of 
accuracy  required  calls  for  good  judgment  on  the 
part  of  the  draftsman.  Limits  may  be  expressed 
in  two  ways.  For  instance,  a  running  fit  on  a 
shaft  to  go  into  a  li  inch  standard  size  hole  may 
be  marked 

-0.0005  max. 
l5-0.0015min. 

or  it  may  be  expressed 

1.4995  max. 
1.4985  min. 

which  means  that  the  shaft  must  not  be  larger 
than  1.4995  inch,  and  not  smaller  than  1.4985  inch. 
On  drawings,  the  tap  drill  size  and  the  depth  of 
tapped  holes  should  always  be  shown.  Surfaces 
to  be  ground  to  size  should  be  marked  " grind/' 
If  the  surface  is  to  be  filed,  the  words  "file  finish' ' 
are  substituted  for  the  letter  "f."  Finishing 
marks,  as  a  rule,  are  used  on  castings  and  forg- 
ings  only.  On  work  made  from  bar  stock,  every 
surface  is  nearly  always  finished,  so  that  here  the 
finishing  marks  are  omitted.  When  a  casting  or 
forging  is  finished  on  every  surface,  it  is  not  nec- 
essary to  show  finish  marks,  but  the  words  "finish 
all  over"  may  be  written  in  a  conspicuous  place, 
so  as  to  readily  catch  the  eye  of  the  workman.  If, 
on  work  made  from  bar  stock,  it  is  desired  that 
the  piece  be  left  rough  at  any  point,  the  words 


WORKING  DRAWINGS  65 

"stock  size"  may  be  applied  to  the  figures  giving 
that  particular  dimension.  For  instance,  on  a 
li-inch  cold  rolled  shaft,  turned  for  journals  for  a 
short  distance  at  each  end,  the  central  part  would 
be  dimensioned  "li-inch  stock  size/' 

While  the  practice  of  indicating  finished  surfaces 
by  the  letter  "f"  is  by  far  the  most  frequently 
met  with,  it  is  by  no  means  universal.  In  some 
shops  the  words  " polish, "  "ream,"  "finish,"  etc., 
are  written  near  the  lines  representing  the  sur- 
faces to  be  thus  treated.  Still  another  method 
much  in  use  is  to  draw  a  red  line  outside  of  the 
line  representing  each  surface  to  be  finished.  If 
a  blue-print  is  made  from  a  tracing  thus  pre- 
pared, the  red  lines  will  print  fainter  than  the 
black  ones,  and  the  finish  lines  on  the  blue-prints 
are  traced  over  with  a  red  pencil  or  red  ink  before 
being  sent  out  in  the  shop.  This  method,  how- 
ever, is  more  expensive  than  that  of  indicating 
the  finished  surfaces  by  the  letter  "f,"  and  on 
complicated  drawings,  the  many  additional  red 
lines  tend  to  cause  confusion.  By  whatever 
method  the  finish  is  indicated,  the  finishing 
marks  should  always  be  shown  fully  in  every  view 
of  the  object. 

It  frequently  happens  that  the  representation  of 
an  object  is  made  clearer  by  the  use  of  sectional 
views,  representing  the  object  as  having  been  cut 
in  two,  either  wholly  or  in  part.  Examples  of  this 
are  shown  in  Figs.  69,  70  and  71.  From  these 
illustrations  it  is  apparent  that  the  construction  of 
the  various  pieces  is  much  more  clearly  exhibited 
when  a  section  is  shown.  The  surface  "cut"  or 


66  SELF-TAUGHT  MECHANICAL  DRAWING 

shown  in  section  is  cross-hatched  or  cross-sectioned 
with  fine  lines  at  a  distance  apart  varying  from  a 
thirty-second  to  an  eighth  of  an  inch,  according 
to  the  size  of  the  drawing  and  the  piece.  The 
cross-sectioning  brings  the  parts  in  section  into 
bold  contrast  with  the  remainder  of  the  drawing, 
and  prevent  all  confusion  as  to  what  parts  are  in 
section  and  what  parts  shown  in  full.  All  lines 
beyond  the  sectional  surface  which  are  exposed  to 
view,  should  be  shown  in  the  drawing  as  usual. 
Should  it  be  deemed  necessary,  which  it  seldom  is, 
to  show  any  parts  that  have  been  cut  away  for  the 
purpose  of  showing  a  section,  such  parts  may  be 
drawn  in  by  dash-dotted  lines,  this  indicating  that 
the  parts  thus  shown  are  in  front  of  the  section 
and  actually  cut  away. 

When  a  mechanism  is  shown  in  section,  the  dif- 
ferent parts  of  the  same  pieces  should  always  be 
cross-sectioned  by  lines  inclined  in  the  same  direc- 
tion, while  separate  pieces  adjoining  each  other 
should  always,  when  possible,  be  cross-sectioned 
by  lines  running  in  different  directions.  When  a 
solid  round  piece  is  exposed  to  view  by  a  section, 
it  is  customary  to  show,  it  solid,  and  not  to  section 
it;  the  screw  stud  in  Fig.  69  is  an  example  of  this 
practice. 

Sectional  views  may  also  be  used  for  many  pur- 
poses where  a  slight  deviation  from  the  theory  of 
projection  will  tend  to  simplify  the  representation 
of  certain  machine  details.  The  shape  of  the  arm 
of  a  pulley  or  gear,  or  of  any  other  part  of  a  cast- 
ing, may  be  conveniently  represented  in  this  way. 
The  cutting  plane  may  be  assumed  to  lie  at  any 


WORKING  DRAWINGS 


67 


angle  necessary  to  bring  out  the  details  most  clear- 
ly. A  sectional  view,  for  instance,  may  represent 
a  casting  as  though  it  were  cut  through  partly  on 
one  plane  and  partly  on  another.  In  all  such  cases, 
however,  it  should  be  indicated  in  another  view  of 
the  object  just  where  the  sectional  views  are  sup- 


SECTION  AT 
G-H 


FIG.  78.— Methods  of  Showing  Sections. 

posed  to  be  taken,  so  that  no  confusion  may  arise 
on  this  account.  The  examples  in  the  following 
will  serve  to  make  clear  the  principles  laid  down. 
In  Fig.  78  are  shown  sections  of  two  hand- 
wheels.  When  an  object  is  symmetrical  it  is 
unnecessary  to  show  more  than  one  half  in  sec- 
tion, although  it  is  quite  common  to  section  gears, 
pulleys,  etc.,  completely  on  working  drawings. 
The  hand-wheel  at  A  in  Fig.  78  is  represented  as 


68 


SELF-TAUGHT  MECHANICAL  DRAWING 


though  cut  in  two  along  its  diameter  BC.  When 
the  section  is  taken  along  the  center  line,  it  is  not 
absolutely  necessary  to  explain  where  the  section 
is  taken ;  but  it  can  do  no  harm  to  make  a  practice 
of  in  all  cases  to  state  where  the  section  is  made, 
except  when  perfectly  obvious.  In  this  case  it 
would  be  clear  that  the  section  is  taken  through 


SECTION  AT  A-B 


FIG.  79.— A  Gear-wheel  in  Section. 

the  center,  and  the  legend  "Section  at  BC'9  is 
given  only  to  show  the  principle.  The  hand-wheel 
at  D  is  provided  with  four  arms,  and  the  method 
of  representing  the  shape  of  the  arms,  hub  and 
rim  are  clearly  indicated. 

In  Fig.  79  are  shown  two  views  of  a  gear-wheel, 
indicating  the  conventional  method  of  represent- 
ing gears  on  drawings.  The  view  on  the  left  side 
is  the  side  view,  and  as  all  the  teeth  are,  of  course, 


WORKING  DRAWINGS  69 

alike,  it  is  unnecessary  to  draw  more  than  a  few 
of  them.  The  pitch  line  of  the  teeth  is  represented 
by  a  dash-dotted  line.  In  the  part  of  the  gear- 
wheel rim  where  the  teeth  are  not  shown,  the  face 
of  the  gear  is  indicated  by  a  solid  line,  and  the 
bottom  of  the  teeth  by  a  dotted  line.  In  the  case 
of  machine-cut  gearing,  where  the  teeth  are  cut 
by  standard  formed  cutters,  it  is  unnecessary  to 
show  any  teeth  at  all  on  the  rim  of  the  gear,  it 
being  sufficient  to  state  the  pitch  and  the  number 
of  teeth,  as  will  be  more  fully  explained  later  in 
the  chapter  on  gearing.  To  show  the  shape  to 
which  the  arms  are  formed,  a  sectional  view  of 
one  of  the  arms  is  drawn  in  the  side  view;  the 
ends  of  the  shaft  are  supposed  to  be  broken  off, 
and  are,  therefore,  sectioned  as  shown.  The  right- 
hand  view  of  the  gear  is  a  section  taken  along  the 
line  AB.  It  will  be  noted  that  the  shaft  and  key 
are  not  sectioned,  usual  practice  being  followed  in 
this  respect.  The  gear  shown  has  five  arms,  and 
the  line  AB  cuts  through  one  of  them  only.  This 
arm,  however,  is  not  sectioned  in  the  right-hand 
view,  and  two  opposite  arms  are  drawn  as  though 
both  of  them  lay  in  the  plane  of  the  paper.  While 
this  is  not  theoretically  correct,  it  is  the  method 
usually  followed  because  of  simplicity  in  drawing 
and  clearness  of  representation.  The  method  of 
representing  the  gear  teeth  in  the  sectional  view 
is  the  one  commonly  employed. 

Sectional  and  top  views  of  a  cylinder  end  with 
flange  and  cover  are  shown  in  Fig.  80.  This 
cylinder  cover  has  only  five  bolts,  and  the  plane 
through  which  the  section  is  taken  cuts  through 


70 


SELF-TAUGHT  MECHANICAL  DRAWING 


only  one  of  the  bolts.  It  is  common  practice,  how- 
ever, to  draw  the  section  as  shown  at  the  left. 
The  bolts  are  shown  as  if  two  of  them  were  in  the 
plane  of  the  section.  The  bolts  are  not  sectioned, 


FIG.  80. — Section  of  Cylinder  End  with  Flange  and  Cover. 

but  are  drawn  in  full,  as  explained  previously. 
Dotted  lines  of  the  remaining  bolts,  or  full  lines 
of  their  nuts,  should  not  be  shown,  because  this 
detracts  from  the  clearness  of  the  drawing;  the 
top  view  shows  clearly  the  number  of  the  bolts 
and  their  arrangement,  and  that  is  all  that  is  nec- 
essary. Some  draftsmen  prefer  to  draw  sections 


WORKING  DRAWINGS 


71 


of  this  kind  as  indicated  at  the  right  in  Fig.  80. 
This  method,  however,  is  not  as  commonly  used. 
In  a  case  where  the  object  is  rather  unsymmet- 
rical,  as,  for  instance,  in  Fig.  81,  the  draftsman's 
judgment  must  often  be  relied  upon  to  decide  how 


|< A -»< A 

FlG.  81. — Another  Method  of  Showing  Sections. 

it  shall  best  be  shown  in  section.  Usually  the 
sectional  view  is  made  symmetrical  as  shown,  the 
distances  A  in  the  lower  view  being  made  equal  to 
the  radius  A  in  the  top  view, 

The  materials  for  the  various  details  making  up 
a  complete  mechanism  are  usually  cross-sectioned 


CF  "HE 

UNIVERSITY 

OF 


72 


SELF-TAUGHT  MECHANICAL  DRAWING 


in  such  a  way  as  to  indicate  the  material  from 
which  each  piece  is  made.  There  is,  however,  no 
universally  adopted  or  recognized  standard  for 
cross-sectioning  for  the  purpose  of  indicating  dif- 
ferent materials.  In  Fig.  82  is  shown  a  chart, 


FIG.  82. — Cross-sectioning  used  for  Indicating  Different 
Materials. 


published  by  Mr.  I.  G.  Bayley  in  Machinery,  Oc- 
tober, 1906,  which  represents  average  practice, 
although  it  must  be  distinctly  understood  that 
there  is  no  agreement  in  all  respects  between  the 
numerable  charts  in  use  in  various  drafting-rooms. 
For  this  reason,  cross-sectioning  alone  should 
never  be  depended  upon  for  indicating  to  the  work- 


WORKING  DRAWINGS 


73 


man  the  kind  of  material  to  be  used.  Written 
directions  should  also  be  given,  the  kind  of  mate- 
rial for  each  part  being  plainly  marked.  Tool  steel 
may  be  abbreviated  "T.  S.",  machine  steel,  "M. 


ROUND  BAR,   SOLID 


ROUND  BAR,   HOLLOW 


SQUARE  OR  RECTANGULAR  BAR 


WOODEN  BEAM 


FIG.  83.— "Broken' 


I-BEAM 

Drawings  of  Long  Objects. 


S.";  wrought  iron,  "W.  L";  cast  iron,  "C.  L", 
etc.  The  less  common  materials  in  machine  con- 
struction, such  as  bronze,  brass,  copper,  etc. ,  should 
preferably  be  written  out  in  full,  in  order  to  avoid 
any  chances  for  confusion.  It  is  better  to  be  too 


74 


SELF-TAUGHT  MECHANICAL  DRAWING 


explicit  as  regards  the  information  on  the  draw- 
ing, than  to  risk  misunderstandings  and  conse- 
quent errors. 

Long  bars,  shafting,  structural  beams,  etc. ,  can- 
not conveniently  be  shown  for  their  full  length  on 
the  drawing.  In  such  cases  the  pieces  are  drawn 
as  long  as  the  drawing  and  the  adopted  scale  per- 
mit, and  are  broken  as  shown  in  Fig.  83,  a  part 
between  the  two  end  portions  shown  being  imag- 
ined as  broken  out.  The  di- 
mensions, of  course,  are  given 
for  the  full  length  of  the  piece, 
as  if  not  broken. 

There  are  several  conven- 
tional methods  for  showing 
screw  threads;  these  methods 
are  adopted  largely  for  saving 
of  time,  as  it  would  be  out  of 
the  question  to  spend  the  time 
required  for  drawing  a  true 
helical  screw  thread  on  a  work- 
ing drawing.  A  method  for 
very  nearly  approximating  the 
appearance  of  a  theoretically  correct  screw  drawing 
is  shown  in  Fig.  84,  where  the  projection  of  the 
screw  helix  is  drawn  by  straight  lines.  The  V- 
shaped  outline  is  first  laid  out,  and  the  connecting 
lines  are  then  drawn.  It  will  be  noticed  that  the 
lines  representing  the  roots  of  the  threads  are  not 
parallel  with  those  representing  the  tops  or  points. 
This  aids  in  making  the  drawing  resemble  that  of 
a  true  helix. 
Usually,  however,  much  simpler  methods  are 


FIG.  84.— Method  of 
Drawing  a  Screw, 
Giving  Correct  He- 
lix Effect. 


WORKING  DRAWINGS 


75 


employed  for  indicating  screw  threads.  In  Fig. 
85,  A,  B  and  C,  some  of  these  methods  are  shown. 
When  a  long  piece  is  threaded  the  entire  length, 
this  fact  can  be  indicated  as  at  D,  which  saves 
drawing  the  conventional  thread  for  the  full  length 
of  the  piece.  The  lines  indicating  the  thread  are 


L.H 


E  F 

FIG.  85. -Simplified  Methods  for  Showing  Screw  Threads. 

inclined,  the  same  as  would  be  the  lines  represent- 
ing the  true  helix.  At  E  in  Fig.  85  is  shown  a 
right-hand  thread  and  at  F  a  left-hand  thread,  the 
different  direction  of  inclination  of  the  thread  in- 
dicating this  fact.  However,  if  a  thread  is  to  be 
left-hand,  it  should  always  be  so  marked  on  the 
drawing.  It  is  usual  to  abbreviate  left-hand,  writ- 
ing "L.  H." 


76 


SELF-TAUGHT  MECHANICAL  DRAWING 


Three  methods  of  indicating  tapped  holes  are 
shown  in  Fig.  86,  these  being  used  when  the  holes 
are  obscured  from  view,  and  shown  by  dotted  lines. 
When  a  tapped  hole  is  shown  in  section,  and  looked 
upon  from  the  top,  it  is  shown  as  indicated  at  D, 
while  if  seen  from  the  side,  in  section,  it  is  repre- 


FIG.  86.— Simplified  Methods  for  Indicating  Tapped  Holes. 

sented  as  at  E.  A  surface  having  tapped  holes  in 
it,  seen  from  above,  is  shown  at  F.  At  G  and  H 
are  shown  the  methods  of  representing  bolts  or 
screws  inserted  in  place  in  tapped  holes.  It  will 
be  noted  that  when  the  threads  of  a  tapped  hole 
are  exposed  to  view  by  section,  the  lines  repre- 
senting the  screw  helix  will  be  seen  to  slope  in  the 
opposite  direction  to  those  of  the  screw,  it  being 


WORKING  DRAWINGS  77 

the  back  side  that  is  exposed  to  view.  An  example 
of  this  is  shown  in  Fig.  71  as  well  as  in  Fig.  86. 

In  drawings  made  for  use  in  the  shop  it  is  cus- 
tomary to  make  the  lines  of  uniform  thickness. 
For  shop  use  such  drawings  are  as  good  as  any. 
When,  however,  the  purpose  of  a  drawing  is  chiefly 
to  show  up  the  object  which  it  represents,  its  ef- 
fectiveness may  be  considerably  enhanced  by  the 
use  of  shade  lines  as  shown  in  Fig.  87.  In  shade 
line  work,  the  light  is  usually  assumed  to  come 
from  the  upper  left  hand  corner, 
and  to  shine  diagonally  across 
the  paper  at  an  angle  of  forty- 
five  degrees.  Lines  on  the  side 
of  the  object  away  from  the 
light,  or  lines  separating  light 
from  dark  surfaces,  are  made 


extra  heavy.     This  gives  to  the      FlG  87>_Use  Of 
drawing  a  suggestion  of  relief.          shade  Lines. 
An  examination  of  the  lines  of 
Fig.  87  taken  in  connection  with  the  direction  from 
which  the  light  is  supposed  to  come  will  show, 
without  the  aid  of  any  other  view,  that  the  hex- 
agonal part  is  raised  above  the  surface  of  the 
square,  and  that  the  circle  in  the  center  represents 
a  depression. 

When  a  drawing  is  intended  for  permanent  use 
it  is  customary  to  make  only  a  pencil  layout  on 
paper,  usually  on  brown  paper,  and  from  this  to 
make  a  tracing  from  which  any  number  of  blue 
print  copies  may  be  made.  The  tracing  is  usually 
made  on  the  regular  tracing  cloth.  This  has  one 
glazed  and  one  unglazed  surface.  Either  surface 


78  SELF-TAUGHT  MECHANICAL  DRAWING 

may  be  used.  The  tracing  cloth  is  drawn  tightly 
over  the  pencil  drawing,  and  its  surface  is  cleaned 
of  any  greasiness  with  dry  powdered  chalk.  This 
insures  a  good  flow  to  the  ink.  In  doing  the  ink 
work  curved  lines  should  be  made  first,  straight 
lines  afterwards,  as  mentioned  in  Chapter  I. 

The  blue  prints  are  made  in  the  same  manner  as 
photographs  are  printed,  the  tracing  taking  the 
place  of  the  photographic  negative.  An  exposure 
of  from  three  to  ten  minutes  may  be  required,  de- 
pending on  the  freshness  of  the  blue  print  paper 
and  the  brightness  of  the  sun.  After  the  proper 
exposure  has  been  given,  which  may  require  some 
experimenting  at  first,  until  one  gets  accustomed 
to  the  change  in  the  paper  which  the  light  makes, 
the  print  is  thoroughly  rinsed  out  in  clear  water 
and  dried,  by  being  hung  up  by  one  edge. 

White  writing  may  be  made  on  a  blue  print  with 
saleratus  water,  the  water  being  given  all  the  sale- 
ratus  it  will  dissolve. 


CHAPTER  VI 

ALGEBRAIC  FORMULAS 

IN  order  to  be  able  to  carry  out  the  calculations 
required  in  simple  machine  design,  it  is  necessary 
that  a  general  understanding  of  the  use  of  for- 
mulas, such  as  are  used  in  mechanical  hand-books 
and  in  articles  in  the  technical  press,  is  acquired. 
Knowledge  of  algebra  or  so-called  "  higher  mathe- 
matics" is  by  no  means  necessary,  although,  of 
course,  such  knowledge  is  very  valuable ;  but  simple 
formulas  can  be  used,  and  the  results  of  scientific 
results  employed  in  practical  work  to  a  very  great 
extent,  by  any  man  who  understands  how  to  use 
the  formulas  given  by  the  various  authorities ;  and 
the  knowledge  required  for  an  intelligent  use  of 
algebraic  formulas  can  be  very  easily  acquired. 
All  the  mathematical  knowledge  necessary  as  a 
foundation  is  a  clear  understanding  of  the  funda- 
mental rules  and  processes  of  arithmetic. 

A  formula  is  simply  a  rule  expressed  in  the  sim- 
plest and  most  compact  manner  possible.  By  using 
letters  and  signs  in  the  formula  instead  of  the 
words  in  the  rule,  it  is  possible  to  condense,  in  a 
very  small  space,  the  essentials  of  long  and  cum- 
bersome rules.  The  letters  used  in  formulas  sim- 
ply stand  in  place  of  the  figures  which  would  be 
used  for  solving  any  specific  problem ;  the  signs 
used  are  the  ordinary  arithmetical  signs  used  in 

79 


80  SELF-TAUGHT  MECHANICAL  DRAWING 

all  kinds  of  calculations.  As  each  letter  stands  for 
a  certain  number  or  quantity,  whenever  a  specific 
problem  is  solved  the  figures  for  that  case  are  put 
into  the  formula  in  place  of  the  letters,  and  the 
calculation  is  carried  out  as  in  ordinary  arithmetic. 
This  may,  perhaps,  be  made  clearer  by  means  of  a 
few  examples. 

The  circumference  of  a  circle  equals  the  diameter 
times  3.1416.  This  rule  may  be  written  as  a 
formula  as  follows : 

C=  DX  3.1416. 

In  this  formula  C  =  circumference,  and  D  = 
diameter.  No  matter  what  the  diameter  is,  this 
formula  says,  the  circumference  is  always  equal  to 
the  diameter  (D)  times  3.1416.  Assume  that  the 
diameter  is  5  inches.  Then,  to  find  the  circumfer- 
ence, place  5  in  the  formula  in  place  of  D. 

C  =  5  X  3.1416  =  15.708  inches. 
If  the  diameter  of  a  circle  is  12  feet,  then 
C  =  12  X  3.1416=37.6992  feet. 

This,  of  course,  is  the  very  simplest  kind  of  a 
formula,  but  it  illustrates  the  principle  involved, 
and  indicates  how  easily  formulas  may  be  em- 
ployed. 

One  of  the  most  well-known  formulas  in  steam 
engineering  is  that  giving  the  horse-power  of  an 
engine,  when  the  average  or  mean  effective  pres- 
sure of  the  steam  on  the  piston,  the  length  of  the 
stroke  of  the  piston  in  feet,  the  area  of  the  piston 
in  square  inches,  and  the  number  of  strokes  per 
minute,  are  known.  Let 


ALGEBRAIC  FORMULAS  81 

H.P.  =  horse-power, 

P  =  mean  effective  pressure  in  pounds  per 

square  inch, 

L  =  length  of  stroke  in  feet, 
A  =  area  of  piston  in  square  inches,  and 
N  =  number  of  strokes  per  minute. 

Then 

PX  LX A  X  N 


H.P. 


33,000 


The  rule  conveying  this  information  expressed 
in  words  would  require  considerable  space,  and  be 
difficult  to  grasp  immediately  ;  but  the  meaning  of 
the  formula  is  quickly  understood.  If  the  pressure 
(P)  equals  75  pounds,  the  stroke  (L)  2  feet,  the 
area  of  the  piston  (A)  125  square  inches,  and  the 
number  of  strokes  per  minute  (N)  60,  then 

TT  D       75  X  2  X  125  X  60     OA 
H'P'  =         ~~ 


It  will  be  seen  that  the  values  for  the  different 
quantities  are  merely  inserted  in  the  formula  in 
place  of  the  corresponding  letters,  and  then  the 
calculation  is  carried  out  as  usual.  It  will  be 
remembered  that  the  line  between  numerator  and 
denominator  in  a  fraction  also  means  a  division; 
that  is 

i   i  OK  f)(\f) 

^  =  1,125,000  -*•  33,000  -  34.1. 


It  is  very  common  in  formulas  to  leave  out,  en- 
tirely, the  sign  of  multiplication  (  X  )  between  the 
letters  expressing  the  values  of  the  various  quanti- 
ties that  are  to  be  multiplied.  Thus,  for  example, 


82  SELF-TAUGHT  MECHANICAL  DRAWING 

PL  means  simply  P  X  L,  and  if  P  =  21  and  L  =  3, 
then  PL  =  P  X  L  =  21  X3  =  63.  If  the  multipli- 
cation signs  are  left  out  in  the  formula  for  the 
horse-power  of  engines  just  referred  to,  the  for- 
mula 

PXLXA  XN       ,,  ,  '        ...      PLAN 
-  could  be  written 


As  a  further  example  of  the  leaving  out  of  the 
multiplication  sign  in  a  formula,  assume  that  D 
=  12,  R  =  3,  and  r  =  2,  then 

DRr       DXRXr  =  12  X  3  X  2       72  _ 
9  9  9  :   9   : 

It  must  be  remembered  that  no  other  signs,  ex- 
cept the  multiplication  sign,  may  thus  be  left  out 
between  the  letters  in  a  formula. 

From  the  examples  given,  the  use  of  simple 
formulas  is  clear;  each  letter  stands  for  a  cer- 
tain number  or  quantity  which  must  be  known  in 
order  to  solve  the  problem  ;  when  the  formula  is 
used  for  the  solution  of  a  problem,  the  letters  are 
simply  replaced  by  the  corresponding  number, 
and  the  result  is  found  by  regular  arithmetical 
operations. 

The  expressions  "square"  and  "square  root" 
and  "cube"  and  "cube  root"  are  frequently  used 
in  engineering  hand-books  and  technical  journals. 
It  would  seem,  to  one  unfamiliar  with  these  names 
and  their  mathematical  meaning,  as  well  as  the 
signs  by  which  they  are  indicated,  that  difficult 
mathematical  operations  are  involved;  but  this  is 
not  necessarily  always  the  case.  The  square  of  a 
number  is  simply  the  product  of  that  number  mul- 


ALGEBRAIC  FORMULAS  83 

tiplied  by  itself.  Thus  the  square  of  3  is  3  X  3  =  9, 
and  the  square  of  5  is  5  X  5  =  25.  In  the  same 
way,  the  square  of  81  is  81  X  81  =  6561.  Instead 
of  writing  81 X  81,  it  is  common  practice  in 
mathematics  to  write  812,  which  is  read  "81 
square/'  and  indicates  that  81  is  to  be  multiplied 
by  itself.  Similarly,  we  may  write  72  =  7  X  7  = 
49,  and  12 2  =  12  X  12  =  144.  The  little  "2"  in  the 
upper  right-hand  corner  of  these  expressions  is 
called  "exponent."  Nearly  all  mechanical  and 
engineering  hand-books  are  provided  with  tables 
which  give  the  squares  (and  also  the  square  root, 
cube  and  cube  root)  of  all  numbers  up  to  1000,  so 
that  it  is  usually  unnecessary  to  calculate  these 
values  by  actual  multiplication. 

As  the  squares  of  numbers  are  frequently  used 
in  formulas  for  solving  problems  occurring  in 
machine  design  and  machine-shop  calculations,  a 
few  examples  will  be  given  below  of  formulas  con- 
taining squares. 

The  area  of  a  circle  equals  the  square  of  the 
radius  multiplied  by  3.1416.  Expressed  as  a  for- 
mula, if  A  =  area  of  circle,  R  =  radius,  and  the 
Greek  letter  n  (Pi)  =  3.1416,  we  have: 

A  =  E2  K. 

If  we  want  to  know  the  area  of  a  circle  having  a 
5-foot  radius,  we  have : 

A  =  527r=5X5X  3.1416  -  78.54  square  feet. 

As  a  further  example,  assume  a  formula  to  be 
given  as  follows : 

A  _  D2N  +  R2n 
A~         DR 


84  SELF-TAUGHT  MECHANICAL  DRAWING 

Assume  that  D  =  3,  N  =  5,  R  =  4,  and  n  (as 
usual)  =3.1416.  What  is  the  value  of  A?  Insert- 
ing the  values  of  the  various  letters  in  the  formula, 
we  have : 

32X  5  +  42X  TT       3X3X5  +  4X4X7T 


A  = 


3X4  3X4 


9  X  5  +  16  X  n  _  45  +  50.2656  _  95J2656  _  _  QQQQ 
12  12  12 

It  will  be  seen  in  the  example  above  that  all  the 
multiplications  are  carried  out  before  any  addition 
is  made.  This  is  in  accordance  with  the  rules  of 
mathematics.  When  several  numbers  or  expres- 
sions are  connected  with  signs  indicating  that 
additions,  subtractions,  multiplications  or  divisions 
are  to  be  made,  the  multiplications  should  be 
carried  out  before  any  of  the  other  operations, 
because  the  numbers  that  are  connected  by  the 
multiplication  sign  are  actually  only  factors  of 
the  product  thus  indicated,  and  consequently  this 
product  must  be  considered  as  one  number  by 
itself.  The  other  operations  are  carried  out  in  the 
order  written,  except  that  divisions  when  written 
in  line  with  additions  and  subtractions,  precede 
these  operations.  A  number  of  examples  of  these 
rules  are  given  below: 

12  X  3  +  7  X  2i  -  li=  36  +  17J  -  1J  =  52. 
5  +  13X7-2=5  +  91-2  =  94.    * 
9-3  +  9X3=3  +  27  =  30. 
9  +  9-3-2=9  +  3-2  =  10. 

Sometimes,  however,  in  formulas,  it  is  desired 
that  certain  operations  in  addition  and  subtraction 


ALGEBRAIC  FORMULAS  85 

precede  the  multiplications.  In  such  cases  use  are 
made  of  the  parenthesis  (  )  and  bracket  [  ].  These 
mathematical  auxiliaries  indicate  that  the  expres- 
sion inside  of  the  parenthesis  or  bracket  should  be 
considered  as  one  single  expression  or  value,  and 
that,  therefore,  the  calculation  inside  the  parenthe- 
sis or  bracket  should  be  carried  out  by  itself  com- 
plete before  the  remaining  calculations  are  com- 
menced. If  one  bracket  is  placed  inside  of  another, 
the  one  inside  is  first  calculated,  and  when  com- 
pleted the  other  one  is  carried  out.  Some  examples 
will  illustrate  these  rules  and  principles: 

(6  -  2)  X  3  +  4  =4  X  3  +  4  =  12+  4  =  16. 

3  X  (12  +  7)  -  28i  =  3  X  19  -  28i  =  57-28J  =2. 

3  +  [5  X  3  (5  +  2)  -  3]  X  6  =  3  +  [5  X  3  X  7  -3] 
X  6  =  3  +  [105  -  3]  X  6  =  3  +  102  X  6  =  3  +  612 
=  615. 

Without  the  parentheses  and  brackets,  the  calcu- 
lations above  would  have  been  as  follows : 

6-2X3  +  4  =  6-6  +  4  =  4. 

3  X  12  +  7  H-  28i  =  36  +  0.2456  =  36.2456. 

3  +  5X3X5  +  2-3X6  =  3  + 75 +  2-  18  =  62. 

These  examples  should  be  carefully  studied  until 
thoroughly  understood. 

We  are  now  ready  to  return  to  the  question  of 
square  roots.  The  square  root  of  a  number  is  that 
number  which,  if  multipled  by  itself,  would  give 
the  given  number.  Thus,  the  square  root  of  9  is  3, 
because  3  multiplied  by  itself  equals  9.  The  square 
root  of  16  equals  4,  of  36  equals  6,  and  so  forth.  It 
will  be  seen  at  once  that  the  square  root  may  be 


86  SELF-TAUGHT  MECHANICAL  DRAWING 

considered,  or,  rather,  actually  is  the  reverse  of 
the  square,  so  that  if  the  square  of  20  is  400,  then 
the  square  root  of  400  is  20.  In  the  same  way,  as 
the  square  of  100  is  10,000,  so  the  square  root  of 
10,000  is  .100.  The  sign  used_m  mathematical 
formulas  for  the  square  root  is  V  .  Thus  V  9  =  3, 
V  49  =  7,  and  .  so  forth.  The  process  of  actually 
calculating  the  square  root  is  rather  cumbersome, 
and  it  is  very  seldom  required,  because,  as  already 
mentioned,  the  engineering  hand-books  usually 
give  tables  of  square  roots  for  all  numbers  up  to 
1000,  and  for  larger  numbers  the  tables  can  also  be 
used  for  obtaining  the  square  root  approximately 
correct,  or  at  least  near  enough  so  for  almost  all 
practical  calculations. 

The  cube  of  a  number  is  the  product  resulting 
from  repeating  the  given  number  as  a  factor  three 
times.  Thus,  the  cube  of  3  is  3  X  3  X  3  -  27,  and 
the  cube  of  17  is  17  X  17  X  17  =  4913.  In  the  same 
way  as  we  write  22  =  2X2  =  4,  for  the  square  of 
2,  so  we  can  write  23  =  2  X2  X  2  =  8,  for  the  cube 
of  2.  The  exponent  (3)  indicates  how  many  times 
the  given  number  is  to  be  repeated  as  a  factor. 
The  cube  of  4,  for  example,  may  be  written  43  =  4 
X  4  X  4  =  64.  Similarly  17 3  -  4913.  The  expres- 
sion 17 3  may  be  read  " the  cube  of  17,"  "17  cube/' 
or  "the  third  power  of  17."  In  the  same  way  as 
the  square  root  means  the  reverse  of  square,  so  the 
cube  root  (or  "third  root")  means  the  reverse  of 
cube  or  "third  power" ;  that  is,  the  cube  root  of  a 
number  is  the  number  which,  if  repeated  as  factor 
three  times,  would  give  the  given  number.  For 
example,  the  cube  root  of  64  is  4,  because  4  X  4  X 


ALGEBRAIC  FORMULAS  87 

4  =  64.  It  is  evident  that  if  the  cube  of  a  number, 
say  6,  is  216  (6  X  6  X  6  =  216),  then  the  cube  root 
of  216  is  6.  The  sign  used  injformulas  for  the  cube 
root  is  f^.  For_example,  f7  8  =  2  (because  2X2 
X  2=  8),  and  1^125  =  5  (because  5X5X5=  125). 
Similarly,  1^3,723,875  =  155. 

The  use  of  the  square  and  square  root,  and  cube 
and  cube  root  in  formulas  may  be  shown  by  a  few 
examples  : 

V  B  X  V  C 


Assume  that  B  =  27,  C  =  25,  and  D  =  2.     Insert 
these  values  in  the  formula.     Then 


X  125         3X5          15 

252  +  22      "  125  +4  ""   129  " 


As  another  example: 
z? 
A 


B3  X  V  C 

Assume  B  =  2,  C  =  9,  and  D  =  4.  Then 

_  22  +  92+42  _  4  +  81  +  16  _.  101. 

23XT/y  8X3  24 

In  the  same  way  as  22  =  2  X  2  =  4,  so  24  =  2  X  2 
X  2  X  2  =  16,  and  25  =  2X2X2X2X2  =  32. 

The  expression  24  is  read  the  "fourth  power  of 
2,"  and  2 5  the  "fifth  power  of  2."  The  exponents 
(4)  and  (5)  indicate  how  many  times  the  given 
number  is  to  be  repeated  as  factor. 

If,  again,  it  is  required  to  find  the  number  which, 
if  repeated  as  factor  four  times,  gives  the  given 
number,  we  must  obtain  the  "fourth  root"  or  V~ 


88  SELF-TAUGHT  MECHANICAL  DRAWING 

Thus,  Vl6  =  2,  ^because  2  X  2  X  2  X  2  =  16.  In 
the  same  way  V256  =  4.  The  fifth  root  is  writ- 
ten V  ;  and  \/243  =  3,  because  3X3X  3X3X3 
=  243. 

These  explanations,  when  fully  understood,  will 
eliminate  all  difficulties  with  formulas  of  a  simple 
nature,  and  with  such  expressions  as  cube  root, 
exponents,  etc. 

An  important  method  facilitating  the  use  of 
formulas,  is  commonly  known  as  the  transposition 
of  formulas.  A  formula  for  finding  the  horse- 
power which  can  safely  be  transmitted  by  a  gear 
of  a  given  size,  running  at  a  given  speed,  is : 

D  X  NX  PX  FX20Q 


H.P.  = 


126,050 


In  this  formula    H.P.  =  horse-power, 
D  =  pitch  diameter, 
N  =  revolutions  per  minute, 
P  =  circular  pitch  of  gear, 
F  =  width  of  face  of  gear. 

Assume,  for  example,  that  the  pitch  diameter  of 
a  gear  is  31.5  inches,  the  number  of  revolutions 
per  minute  200,  the  circular  pitch  1J  inch,  and  the 
width  of  the  face  3  inches.  Then,  if  these  values 
arc  inserted  in  the  formula,  we  have  : 

31.5  X  200  X  lj  X  3  X  200       ,c 
H'R  -  ~  =  45 


power,  very  nearly. 

Assume,  however,  that  the  horse-power  required 
to  be  transmitted  is  known,  and  that  the  pitch  of 
the  gear  is  required  to  be  found.  Assume  that 


ALGEBRAIC  FORMULAS  89 

tf.P  =  30;  Z>  =  31.5;  N  =  200;  F=3;  and  that  P 
is  the  unknown   quantity;    then,     inserting  the 
known  values  in  the  formula,  gives  us: 
31.5  X  200  XPX  3  X  200 
126,050 

In  order  to  be  able  to  find  P,  we  want  it  given 
on  one  side  of  the  equals  sign,  with  all  the  known 
quantities  on  the  other  side.  If  we  multiply  the 
expressions  on  both  sides  of  the  equals  sign  by 
the  same  number  we  do  not  change  the  conditions ; 
thus 
on  v  i oa  AKA  -  31.5  X  200  X  P  X  3  X  200  X  126,050 

1Zb'Ut  126,050 

By  canceling  the  number  126,050  on  the  right- 
hand  side  we  have : 

30  X  126,050  =  31.5  X  200  X  P  X  3  X  200. 
If  we  now  divide  on  both  sides  of  the  equals 
sign  with  31.5  X  200  X  3  X  200,  we  have: 

30  X  126,050          =  31.5X200  XPX  3X200 
31.5  X  200  X  3  X  200          31.5  X  200  X  3  X  200 

We  can  now  cancel  all  numerical  values  in  the 
fraction  on  the  right-hand  side;  then: 
30  X  126,050  p 

31.5X200X3X200 

This  is  then  the  transposed  formula  giving  P, 
and  from  this  we  find  that  P  =  1  inch. 
In  general,  any  formula  of  the  form 

B 

A---C. 

can  be  transposed  as  below : 

A  XC  =  B     C  =      - 


90  SELF-TAUGHT  MECHANICAL  DRAWING 

It  will  be  seen  that  the  quantities  which  are  in 
the  denominator  on  one  side  of  the  equals  sign,  are 
transposed  into  the  numerator  on  the  other  side, 
and  vice  versa. 

Examples: 

BX  C 

A  D      . 

Then: 

n   _  Bx c     D       AX  D     n       AX  D 
A     '       =         C     '       =     ~~B     ' 

A  _  EXFX  G 
KXL 

Then: 
E_A XKXL    F_A XKXL . r  _A X KxL 

FX G  '    EX G  '    EXF  ' 

rf  =  EX  FX  G  T  EXFX  G 
AXL      AXK 

The  principles  of  transposition  of  formulas  can 
best  be  grasped  by  a  careful  study  of  the  examples 
given.  Note  that  the  method  is  only  directly  ap- 
plicable when  all  the  quantities  in  the  numerator 
and  denominator  are  factors  of  a  product.  If  con- 
nected by  +  or  -  signs,  the  transposition  cannot 
be  made  by  the  simple  methods  shown  unless  the 
whole  sum  or  difference  is  transposed.  Example : 


A  =-  ;  then  D  =~and£  +  C=  A  X  D. 

The  most  usual  caclulations,  perhaps,  in  some 
classes  of  machine  design,  are  those  involving  the 
finding  of  the  strength  of  certain  machine  mem- 
bers ;  and,  in  order  to  find  the  strength  qf  these 


ALGEBRAIC  FORMULAS  91 

members,  it  is  necessary  to  first  find  the  cross- 
sectional  area  of  the  part  subjected  to  stress.  For 
this  reason,  the  remainder  of  this  chapter  will  be 
largely  taken  up  with  rules  and  formulas  for  find- 
ing the  areas  and  other  properties  of  various  geo- 
metrical figures.  Rules  and  formulas  for  volumes 
of  solids  will  also  be  given.  Examples  have  been 
given  in  some  cases  merely  to  show  the  applica- 
tions of  the  formulas. 

The  area  of  a  triangle  equals  one-half  the  prod- 
uct of  its  base  and  its  altitude.  The  base  may  be 
any  side  of  the  triangle,  and  the  altitude  is  the 
length  of  the  line  drawn  from  the  angle  opposite 
the  base,  perpendicular  to  it. 

Assume  that  A  =  area  of  triangle, 
.  B  =  base, 
-  H  =  altitude. 

Then  the  rule  above  may  be  expressed  as  a 
formula  as  follows  : 

A  ^BXH 

' 


Let  the  base  (B)  of  a  triangle  be  5  feet,  and  the 
altitude  (H)  8  feet.  Then  the  area 

5X8       40     '  . 

A  =  —  n  —  =  -JT  =  20  square  feet. 

z          z 

The  area  of  a  square  equals  the  square  of  its 
side.  If  A  =  the  area,  and  S  the  side  of  the  square, 
then 


If  the  side  is  9.7  inches  long,  then 

A  =  9.7  2=  9.7  X  9.7  =94.09  square  inches. 


92          SELF-TAUGHT  MECHANICAL  DRAWING 

The  area  of  a  rectangle  equals  the  product  of  its 
long  and  short  sides.  If  A  =  area,  L  =  length  of 
the  longer  side,  and  H=  length  of  the  shorter  side, 
then 

A  =  L  X  H. 

The  area  of  a  parallelogram  equals  the  product 
of  the  base  and  the  altitude. 

The  area  of  a  trapezoid  equals  one-half  the  sum 
of  the  parallel  sides  multiplied  by  the  altitude. 
If  A  =  area,  B  =  length  of  one  of  the  parallel  sides, 
C  =  length  of  the  other  parallel  side,  and  H  = 
altitude,  then 

•          B 


Assume  that  the  lengths  of  the  two  parallel 
sides  are  12  and  9  feet,  respectively,  and  that  the 
altitude  is  16  feet.  Then 

A  =   ^-^  X  16  =  10.5  X  16  =  168  square  feet. 

To  find  the  area  of  an  irregular  figure  bounded 
by  straight  lines,  divide  the  figure  into  triangles, 
and  find  the  area  of  each  triangle  separately.  The 
sum  of  the  areas  of  all  the  triangles  equals  the 
area  of  the  figure. 

The  circumference  of  a  circle  equals  its  diameter 
multiplied  by  3.1416. 

The  diameter  of  a  circle  equals  the  circumfer- 
ence divided  by  3.1416. 

The  area  of  a  circle  equals  the  square  of  the 
diameter  multiplied  by  0.7854. 

The  diameter  of  a  circle  equals  the  area  divided 


ALGEBRAIC  FORMULAS  93 

by  0.7854,  and  the  square   root  extracted  of  the 
quotient. 

If  D  =  diameter,  C  =  circumference,  and  A  = 
area,  these  last  rules  may  be  expressed  in  formulas 
as  follows : 

C-DX  3.1416.         £=3ib- 

A  =  D*X  0.7854.  D  = 


The  length  of  a  circular  arc  equals  the  circum- 
ference of  the  circle,  multiplied  by  the  number  of 
degrees  in  the  arc,  divided  by  360.  If  L  =  length 
of  arc,  C  =  circumference  of  circle,  and  N  =  num- 
ber of  degrees  in  the  arc,  then 

r        CXN 
360 

The  area  of  a  circular  sector  equals  the  area  of 
the  whole  circle  multiplied  by  the  quotient  of  the 
number  of  degrees  in  the  arc  of  the  sector  divided 
by  360.  If  a  =  area  of  sector,  A  =  area  of  circle, 
and  N  =  number  of  degrees  in  sector,  then 

a  =  A  X    N 


360* 

The  area  of  a  circular  segment  equals  the  area  of 
the  circular  sector  formed  by  drawing  radii  from 
the  center  of  the  circle  to  the  extremities  of  the 
arc  of  the  segment,  minus  the  area  of  the  triangle 
formed  by  these  radii  and  the  chord  of  the  arc  of 
the  segment. 

The  area  of  a  pentagon  (regular  polygon  having 


94  SELF-TAUGHT  MECHANICAL  DRAWING 

five  sides)  equals  the  square  of  the  side  times 
1.720. 

The  area  of  a  hexagon  (regular  polygon  having 
six  sides)  equals  the  square  of  the  side  times 
2.598. 

The  area  of  a  heptagon  (regular  polygon  having 
seven  sides)  equals  the  square  of  the  side  times 
3.634. 

The  area  of  an  octagon  (regular  polygon  having 
eight  sides)  equals  the  square  of  the  side  times 
4.828. 

The  volume  of  a  cube  equals  the  cube  of  the 
length  of  its  side. 

The  volume  of  a  prism  equals  the  area  of  the 
base  multiplied  by  the  altitude. 

The  volume  of  a  cylinder  equals  the  area  of  its 
base  circle  multiplied  by  the  altitude. 

The  volume  of  a  pyramid  or  cone  equals  the  area 
of  the  base  times  one-third  the  altitude. 

The  area  of  the  surface  of  a  sphere  equals  the 
square  of  the  diameter  multiplied  by  3.1416. 

The  volume  of  a  sphere  equals  the  cube  of  the 
diameter  times  0.5236. 

The  volume  of  a  spherical  sector  equals  two- 
thirds  of  the  square  of  the  radius  of  the  sphere 
multiplied  by  the  height  of  the  contained  spherical 
segment,  multiplied  by  3.1416.  If  V  =  volume  of 
sector,  R  =  radius  of  sphere,  and  H=  height  of  the 
contained  spherical  segment,  then 

V  =  f-tf2  X  HX  3.1416. 
o 

Assume  that  the  length  of  the  radius  of  a  spheri- 


ALGEBRAIC  FORMULAS  95 

cal  sector  is  6  inches,  and  the  height  of  the  con- 
tained segment  2  inches.     Then 

V  =  -f-X  62  X  2  X  3.1416  =  150.  7968  cubic  inches. 

o 

The  volume  of  a  spherical  segment  equals  the 
radius  of  the  sphere  less  one-third  the  height  of 
the  segment,  multiplied  by  the  square  of  the 
height  of  the  segment,  multiplied  by  3.1416.  If  R 
=  radius,  H  =  height,  and  V  =  volume  of  segment, 
then 


(R-   y) 


XH2  X  3.1416. 


Assume  that  the  length  of  the  radius  is  4  inches, 
and  the  height  of  the  segment  3  inches.  Then 

V  =  (4  -  y)  X  32  X  3.1416  =  84.  8232  cubic  inches. 

The  area  of  an  ellipse  equals  the  long  axis  multi- 
plied by  the  short  axis,  multiplied  by  0.7854.  If 
the  area  =A,  the  long  axis  =B,  and  the  short  axis 
=  C,  then 

A  =  BXCX  0.7854. 

If  the  long  axis  is  12  inches  and  the  short  axis 
8J  inches,  then 

A  =  12  X  8i  X  0.7854  -  78.54. 

Formulas  and  application  of  formulas  have  not 
been  given  for  such  rules  which  are  so  simple  and 
easy  to  understand  that  the  reader  without  diffi- 
culty can  formulate  his  own  formula. 


CHAPTER  VII 

ELEMENTS  OF  TRIGONOMETRY 

TRIGONOMETRY  is  a  very  important  part  of  the 
science  of  mathematics,  and  deals  with  the  deter- 
mination of  angles  and  the  solution  of  triangles. 
In  order  to  fully  understand  the  subjects  treated 
of  in  the  following,  it  is  necessary  that  the  reader 
is  fully  familiar  with  the  usual  methods  of  desig- 
nating the  measurements  or  sizes  of  angles.  While 
mathematicians  employ  also  another  method,  in 
mechanics  angles  are  measured  in  degrees  and 
subdivisions  of  a  degree,  called  minutes.  The 
minute  is  again  subdivided  into  seconds,  but  these 
latter  subdivisions  are  so  small  as  to  permit  of 
being  disregarded  in  general  practical  machine 
design. 

A  degree  is  1-360  part  of  a  circle,  or,  in  other 
words,  if  the  circumference  of  a  circle  is  divided 
into  360  parts,  then  each  part  is  called  one  degree. 

If  two  lines  are  drawn  from  the  center  of  the 
circle  to  the  ends  of  the  small  circular  arc  which 
is  1-360  part  of  the  circumference,  then  the  angle 
between  these  two  lines  is  a  1-degree  angle.  A 
quarter  of  a  circle  or  a  90-degree  angle  is  called  a 
right  angle.  The  meaning  of  obtuse  and  acute 
angles  has  already  been  explained  in  Chapter  II. 
Any  angle  which  is  not  a  right  angle  is  called  an 
oblique  angle. 

96 


ELEMENTS  OF  TRIGONOMETRY  97 

A  minute  is  1-60  part  of  a  degree,  and  a  second 
1-60  part  of  a  minute.  In  other  words,  one  circle 
=  360  degrees,  one  degree  =  60  minutes,  and  one 
minute  =  60  seconds.  The  sign  (°)  is  used  for  in- 
dicating degrees;  the  sign  (')  indicates  minutes, 
and  the  sign  ( " )  seconds.  A  common  abbreviation 
for  degree  is ' l  deg. ' ' ;  for  minute, ' ' min. ' ' ;  and  for 
second,  "sec." 

Two  angles  are  equal  when  the  number  of  de- 
grees they  contain  is  the  same.  If  two  angles  are 
both  30  degrees,  they  are  equal,  no  matter  how 
long  the  sides  of  the  one  may  be  in  relation  to  the 
other. 

Of  all  triangles,  the  right-angled  triangle  occurs 
most  frequently  in  machine  design.  A  right-ang- 
led triangle  is  one  having  the  angle  between  two 
sides  a  right  angle;  the  angles  between  the  other 
sides  may  be  of  any  size.  In  the  calculations  in- 
volved in  solving  right-angled  triangles,  a  useful 
application  of  the  squares  and  square  roots  of 
numbers  is  also  presented.  Assume  that  the  lengths 
of  the  sides  of  a  right-angled  triangle,  as  shown 
in  Fig.  88,  are  5  inches,  4  inches,  and  3  inches, 
respectively.  Then 

52  =  42  +  32,  or  25  =  16  4- 9. 

This  relationship  between  the  three  sides  in  a 
right-angled  triangle  holds  good  for  all  right-ang- 
led triangles.  The  square  of  the  side  opposite  the 
right  angle  equals  the  sum  of  the  squares  of  the 
sides  including  the  right  angle.  Assume,  for  ex- 
ample, that  the  lengths  of  the  two  sides  including 
the  right  angle  in  a  right-angled  triangle  are  12 


98 


SELF-TAUGHT  MECHANICAL  DRAWING 


and  9  inches  long,  respectively,  as  shown  in  Fig. 
89,  and  that  the  side  opposite  the  right  angle,  the 
hypotenuse,  is  to  be  found.  We  then  first  square 
the  two  given  sides,  and  from  our  rule,  just  given, 
we  have  that  the  sum  of  the  squares  equals  the 
square  of  the  side  to  be  found.  The  square  root 


-*• H       h •' 


of  the  sum  must  then  equal  the  side,  itself.    Carry- 
ing out  this  calculation  we  have: 

12  M-_92  =  144  +  81  =  225 

V  225  =  15  inches  =  length  of  hypotenuse. 

Similar  methods  may  be  employed  for  finding 
any  of  the  sides  in  a  right-angled  triangle  if  two 
sides  are  given.  If  the  hypotenuse  were  known 
to  be  15  inches,  and  one  of  the  sides  including  the 
right  angle  9  inches,  as  shown  at  D  in  Fig.  90, 
then  the  other  side  including  the  right  angle  can 
be  found.  In  this  case,  however,  we  must  subtract 
the  square  of  the  known  side  including  the  right 


ELEMENTS  OF  TRIGONOMETRY 


99 


angle  from  the  square  of  the  hypotenuse  to  obtain 
the  square  of  the  remaining  including  side.  We, 
therefore,  have: 

152-92  =  225-81  = 


\/144  =  12  inches  =  length  of  unknown  side. 
In  the  same  way,  if  the  lengths  15  and  12  were 


k; SIDE  TO  BE  FOUND > 


FIG.  90. 


known,  we  could  find  the  side  AC,  as  shown  at  E, 

Fig.  90: 

15^-  122  =  225-  144  =  81 
Vgf  =  9  inches  =  length  of  AC. 

From  these  examples  we  may  formulate  rules 
and  general  formulas  for  the  solution  of  right- 
angled  triangles  when  two  sides  are  known.  In 
Fig.  91,  at  F,  the  square  of  AB  plus  the  square  of 
AC  equals  the  square  of  EC;  the  square  of  EC 
minus  the  square  of  AC  equals  the  square  of  AB; 
and  the  square  of  EC  minus  the  square  of  AB 


100 


SELF-TAUGHT  MECHANICAL  DRAWING 


equals  the  square  of  AC.     These  rules  written  as 
general  formulas  would  take  the  form  : 


BC2-AB2  = 

From  these  formulas  we  have,  by  extracting  the 
square  root  on  each  side  of  the  equal  sign  : 

EC  =  V~AB2  +  AC2 
AB  =  VBC2  -  AC2 
AC  =  V  BC2  -  AB2 

These  formulas  make  it  possible  to  find  the  third 
side  when  two  sides  are  given,  no  matter  what  the 


numerical  values  of  the  length  of  the  sides  may 
be.  Assume  AB  =  12,  and  BC  =  20;  find  AC.  Ac- 
cording to  the  formula : 

AC  =  \/202-  122  =  V400-  144  =  V^56  =16. 

Assume  that  AB  =  15  and  AC  =  20.     Find  BC. 

BC   =Vl52  +  202  =  V  225  +  400  =  \/625  =  25. 

The  rules  and  formulas  given  make  it  possible  to 
find  the  length  of  the  sides  in  a  right-angled  tri- 
angle. To -find  the  angles,  however,  use  must  be 


ELEMENTS  OF  TRIGONOMETRY  101 

made  of  the  trigonometric  functions,  the  meanings 
of  which  will  be  presently  explained.  The  trigo- 
nometric functions  are  the  sine,  cosine,  tangent,  co- 
tangent, secant  and  cosecant  of  angles.  While  these 
functions  are  used  in  the  solution  of  all  kinds  of 
triangles,  they  refer  directly  to  right-angled  tri- 
angles, and  the  meaning  or  value  of  each  function 
can  be  explained  by  reference  to  a  right-angled 
triangle  as  shown  in  Fig.  91,  at  G,  where  the  side 
BC  is  the  hypotenuse,  AC  the  side  adjacent  to 
angle  D,  and  AB  the  side  opposite  angle  D.  Of 
course,  if  reference  is  made  to  angle  E,  then  AB 
is  the  side  adjacent  and  AC  the  side  opposite. 

The  sine  of  an  angle  is  the  length  of  the  opposite 
side,  if  the  hypotenuse  is  assumed  to  equal  1.  The 
sine  of  angle  D,  then,  is  the  length  of  AB  if  BC 
equals  1.  To  find  the  sine  of  D  when  BC  is  any 
other  length,  divide  AB  by  the  length  of  BC.  To 
find  the  sine  of  D,  if  BC  equals  5,  for  example,  it 
is  necessary  to  divide  the  length  of  AB  by  5. 

Find  the  sine  of  D,  when  AB  =  15  and  BC  =  20. 
The  sine  of  D  =  15  *  20  =  0.75. 

The  cosine  of  an  angle  is  the  length  of  the  adja- 
cent side,  if  the  hypotenuse  is  assumed  to  equal  1. 
The  cosine  of  angle  D,  then,  is  the  length  of  AC 
if  BC  equals  1.  To  find  the  cosine  of  D  when  BC 
is  any  other  length,  divide  -AC  by  the  length  of 
BC.  To  find  the  cosine  of  D,  if  BC  equals  8,  for 
example,  it  is  necessary  to  divide  the  length  of 
AC  by  8. 

Find  the  cosine  of  D,  when  AC  =  12  and  BC  = 
30.  The  cosine  of  D  =  12  -*-  30  =  0.4. 

The  tangent  of  an  angle  is  the  length  of  th< 


102         SELF-TAUGHT  MECHANICAL  DRAWING 

posite  side,  if  the  adjacent  side  is  assumed  to 
equal  1.  The  tangent  of  angle  D  is  the  length  of 
AB  if  AC  equals  1.  To  find  the  tangent  of  D  when 
AC  equals  any  other  length,  divide  AB  by  the 
length  of  AC.  To  find  the  tangent  of  D  when  AC 
equals  3,  for  example,  it  is  necessary  to  divide  the 
length  of  AB  by  3. 

Find  the  tangent  of  D,  when  AB  =  16  and  AC 
=  12.  The  tangent  of  D  =  16  +  12  =  1.333. 

The  cotangent  of  an  angle  is  the  length  of  the 
adjacent  side,  if  the  opposite  side  is  assumed  to 
equal  1.  The  cotangent  of  angle  D  is  the  length 
of  AC  if  AB  equals  1.  To  find  the  cotangent  of  D 
when  AB  equals  any  other  length,  divide  AC  by 
the  length  of  AB.  To  find  the  cotangent  of  D 
when  AB  equals  12,  for  example,  divide  AC  by  12. 

Find  the  cotangent  of  D  when  AB  =  3  and  AC 
=  36.  The  cotangent  of  D  =  36  +  3  =  12. 

The  secant  of  an  angle  is  the  length  of  the  hypo- 
tenuse, if  the  adjacent  side  is  assumed  to  equal  1. 
The  secant  of  angle  D  is  the  length  of  BC  when 
AC  equals  1.  To  find  the  secant  of  -D  when  AC  is 
any  other  length,  divide  BC  by  the  length  of  AC. 

Find  the  secant  of  D  when  BC  =  24  and  AC  =  9. 
The  secant  of  D  =  24  -*-  9  =  2.666.  .  . 

The  cosecant  of  an  angle  is  the  length  of  the 
hypotenuse  if  the  opposite  side  is  assumed  to  equal 
1.  The  cosecant  of  angle  D  is  the  length  of  BC 
when  AB  equals  1.  To  find  the  cosecant  of  D  when 
AB  is  any  other  length,  divide  BC  by  the  length 
of  AB. 

Find  the  cosecant  of  D  when  BC  =  30  and  AB 
=  3.75.  The  cosecant  of  D  =  30  ^  3.75  =  8. 


ELEMENTS  OF  TRIGONOMETRY  103 

The  expressions  sine,  cosine,  tangent,  cotangent, 
secant  and  cosecant  are  abbreviated  as  follows: 
sin,  cos,  tan,  cot,  sec,  and  cosec.  Instead  of  writ- 
ing tangent  of  D,  for  example,  it  is  usual  to  write 
tan  D.  By  means  of  these  functions,  tables  of 
which  are  given  in  the  following,  the  values  of 
angles  can  be  introduced  in  the  calculations  of  tri- 
angles. The  tables  here  used  give  the  values  of 
the  functions  of  angles  for  every  degree  and  for 
every  ten  minutes.  Only  three  decimal  places  are 
given,  as  that  is  enough  for  the  great  majority  of 
shop  calculations.  When  very  accurate  calculations 
are  required,  tables  can  be  procured  giving  the 
functions  for  every  minute,  and  with  five  decimal 
places.  From  the  tables  given,  when  the  angle  is 
known,  the  corresponding  angular  function  can  be 
found,  and  when  the  function  is  known,  the  cor- 
responding angle  can  be  determined  by  merely 
reading  off  the  values  in  the  table.  The  tables  in- 
clude sines,  cosines,  tangents  and  cotangents  only, 
as  these  are  most  commonly  used,  and  all  problems 
can  be  solved  by  the  use  of  them.  When  the  se- 
cant is  required,  it  can  be  found  by  dividing  1  by 
the  cosine.  The  cosecant  is  found  by  dividing  1 
by  the  sine. 

The  tables  of  sines,  cosines,  etc.,  are  read  the 
same  as  any  other  table.  It  will  be  seen  that  the 
four  tables  given  are  headed  Sines,  Cosines,  Tan- 
gents, and  Cotangents,  respectively.  At  the  bottom 
of  the  table  headed  "Sines"  is  read  the  word 
"Cosines,"  and  at  the  bottom  of  the  table  headed 
"Cosines"  is  read  the  word  "Sines."  In  the  same 
way,  at  the  bottom  of  the  table  headed  "Tan- 


104 


SINES 


MINUTES. 

DEG. 

DEG. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

0 

0.000 

0.003 

0.006 

0.009 

0.012 

0.015 

0.017 

89 

1 

0.017 

0.020 

0.023 

0.026 

0.029 

0.032 

0.035 

88 

2 

0.035 

0.038 

0.041 

0.044 

0.047 

0.049 

0.052 

87 

3 

0.052 

0.055 

0.058 

0.061 

0.064 

0.067 

0.070 

86 

4 

0.070 

0.073 

0.076 

0.078 

0.081 

0.084 

0.087 

85 

5 

0.087 

0.090 

0.093 

0.096 

0.099 

0.102 

0.105 

84 

6 

0.105 

0.107 

0.110 

0.113 

0.116 

0.119 

0.122 

83 

7 

0.122 

0.125 

0.128 

0.131 

0.133 

0.136 

0.139 

82 

8 

0.139 

0.142 

0.145 

0.148 

0.151 

0.154 

0.156 

81 

9 

0.156 

0.159 

0.162 

0.165 

0.168 

0.171 

0.174 

80 

10 

0.174 

0.177 

0.179 

0.182 

0.185 

0.188 

0.191 

79 

11 

0.191 

0.194 

0.197 

0.199 

0.202 

0.205 

0.208 

78 

12 

0.208 

0.211 

0.214 

0.216 

0.219 

0.222 

0.225 

"77 

13 

0.225 

0.228 

0.231 

0.233 

0.236 

0.239 

0.242 

76 

14 

0.242 

0.245 

0.248 

0.250 

0.253 

0.256 

0.259 

75 

15 

0.259 

0.262 

0.264 

0.267 

0.270 

0.273 

0.276 

74 

16 

0.276 

0.278 

0.281 

0.284 

0.287 

0.290 

0.292 

73 

17 

0.292 

0.295 

0.298 

0.301 

0.303 

0.306 

0.309 

72 

18 

0.309 

0.312 

0.315 

0.317 

0.320 

0.323 

0.326 

71 

19 

0.326 

0.328 

0.331 

0.334 

0.337 

0.339 

0.342 

70 

20 

0.342 

0.345 

0.347 

0.350 

0.353 

0.35) 

0.358 

69 

21 

0.358 

0.361 

0.364 

0.367 

0.369 

0.372 

0.375 

68 

22 

0.375 

0.377 

0.380 

0.383 

0.385 

0.388 

0.391 

67 

23 

0.391 

0.393 

0.396 

0.399 

0.401 

0.404 

0.407 

66 

24 

0.407 

0.409 

0.412 

0.415 

0.417 

0.420 

0.423 

65 

25 

0.423" 

0.425 

0.428 

0.431 

0.433 

0.436 

0.438 

64 

26 

0.438 

0.441 

0.444 

0.446 

0.449 

0.451 

0.454 

63 

27 

0.454 

0.457 

0.459 

0.482 

0.464 

0.467 

0.469 

62 

28 

0.469 

0.472 

0.475 

0.477 

0.480 

0.482 

0.485 

61 

29 

0.485 

0.487 

0.490 

0.492 

0.495 

0.497 

0.500 

60 

30 

0.500 

0.503 

0.505 

0.508 

0.510 

0,513 

0.515 

59 

31 

0.515 

0.518 

0.520 

0.522 

0.525 

0.527 

0.530 

58 

32 

0.530 

0.532 

0.535 

0.537 

0.540 

0.542 

0.545 

57 

33 

0.545 

0.547 

0.553 

0.552 

0.554 

0.557 

0.559 

56 

34 

0.559 

0.562 

0.564 

0.566 

0.569 

0.571 

0.574 

55 

35 

0.574 

0.576 

0.578 

0.581 

0.583 

0.585 

0.588 

54 

36 

0.588 

0.590 

0.592 

0.595 

0.597 

0.599 

0.602 

53 

37 

0.602 

0.604 

0.606 

0.609 

0.611 

0.613 

0.616 

52 

38 

0.616 

0.618 

0.620 

0.623 

0.625 

0.627 

0.629 

51 

39 

0.629 

0.632 

0.634 

0.636 

0.638 

0.641 

0.643 

50 

40 

0.643 

0.645 

0.647 

0.649 

0.652 

0.654 

0.656 

49 

41 

0.656 

0.658 

0.660 

0.663 

0.665 

0.667 

0.669 

48 

42 

0.669 

0.671 

0.673 

0.676 

0.678 

0.680 

0.682 

47 

43 

0.682 

0.684 

0.686 

0.688 

0.690 

0.693 

0.695 

46 

44 

0.695 

0.697 

0.699 

0.701 

0.703 

0.705 

0.707 

45 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

MINUTES. 

COSINES 


COSINES 


105 


MINUTES. 

DEC. 

DEG. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

0 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

~~89~~ 

1 

1.000 

1.000 

1.000 

1.000 

1.000 

0.999 

0.999 

88 

2 

0.999 

0.999 

0.999 

0.999 

0.999 

0.999 

0.999 

87 

3 

0.999 

0.998 

0.998 

0.998 

0.998 

0.998 

0.998 

86 

4 

0.998 

0.997 

0.997 

0.997 

0.997 

0.996 

0.996 

85 

5 

0.996 

0.996 

0.993 

0.995 

0.995 

0.995 

0.995 

84 

6 

0.995 

0.994 

0.994 

0.994 

0.993 

0.993 

0.993 

83 

7 

0.993 

0.992 

0.992 

0.991 

0.991 

0.991 

0.990 

82 

8 

0.990 

0.990 

0.989 

0.989 

0.98) 

0.988 

0.988 

81 

9 

0.988 

0.987 

0.987 

0.986 

0.986 

0.985 

0.985 

80 

10 

0.985 

0.984 

0.984 

0.983 

0.983 

0.982 

0.982 

79 

11 

0.982 

0.981 

0.981 

0.980 

0.979 

0.979 

0.978 

78 

12 

0.978 

0.978 

0.977 

0.976 

0.976 

0.975 

0.974 

77 

13 

0.974 

0.974 

0.973 

0.972 

0.972 

"0.971 

0.970 

76 

14 

0.970 

0.970 

0.969 

0.968 

0.967 

0.967 

0.966 

75 

15 

0.966 

0.965 

0.964 

0.964 

0.963 

0.962 

0.961 

.    74 

16 

0.961 

0.960 

0.960 

0.959 

0.958 

0.957 

0.956 

73 

17 

0.956 

0.955 

0.955 

0.954 

0.953 

0.952 

0.951 

72 

18 

0.951 

0.950 

0.949 

0.948 

0.947 

0.946 

0.946 

71 

19 

0.946 

0.945 

0.944 

0.943 

0.942 

0.941 

0.940 

70 

20 

0.940 

0.939 

0.938 

0.937 

0.936 

0.935 

0.934 

69 

21 

0.934 

0.933 

0.931 

0.930 

0.929 

0.928 

0.927 

68 

22 

0.927 

0.926 

0.925 

0.924 

0.923 

0.922 

0.921 

67 

23 

0.921 

0.919 

0.918 

0.917 

0.916 

0.915 

0.914 

66 

24 

0.914 

0.912 

0.911 

0.910 

0.909 

0.908 

0.906 

65 

25 

0.906 

0.905 

0.904 

0.903 

0.901 

0.900 

0.899 

64 

26 

0.899 

0.898 

0.896 

0.895 

0.894 

0.892 

0.891 

63 

27 

0.891 

0.890 

0.888 

0.887 

0.886 

0.884 

0.883 

62 

28 

0.883 

0.882 

0.880 

0.879 

0.877 

0.876 

0.875 

61 

29 

0.875 

0.873 

0.872 

0.870 

0.869 

0.867 

0.866 

60 

30 

0.866 

0.865 

0.863 

0.862 

0.860 

0.859 

0.857 

59 

31 

0.857 

0.856 

0.854 

0.853 

0.851 

0.850 

0.848 

58 

32 

0.848 

0.847 

0.845 

0.843 

0.842 

0.840 

0.839 

57 

33 

0.839 

0.837 

0.835 

0.834 

0.832 

0.831 

0.829 

56 

34 

0.829 

0.827 

0.826 

0.824 

0.822 

0.821 

0.819 

55 

35 

0.819 

0.817 

0.816 

0.814 

0.812 

0.811 

0.809 

54 

36 

0.809 

0.807 

0.806 

0.804 

0.802 

0.800 

0.799 

53 

37 

0.799 

0.797 

0.795 

0.793 

0.792 

0.790 

0.788 

52 

38 

0.788 

0.786 

0.784 

0.783 

0.781 

0.779 

0.777 

51 

39 

0.777 

0.775 

0.773 

0.772 

0.770 

0.768 

0.766 

50 

40 

0.766 

0.764 

0.762 

0.760 

0.759 

0.757 

0.755 

49 

41 

0.755 

0.753 

0.751 

0.749 

0.747 

0.745 

0.743 

48 

42 

0.743 

0.741 

0.739 

0.737 

0.735 

0.733 

0.731 

47 

43 

0.731 

0.729 

0.727 

0.725 

0.723 

0.721 

0.719 

46 

44 

0.719 

0.717 

0.715 

0.713 

0.711 

0.709 

0.707 

45 

T^FT1 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

J-^-h-O  . 

MINUTES. 

DEG. 

SINES 


106 


TANGENTS 


DEG. 

MINUTES. 

DEG. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

0 

0.000 

0.003 

0.006 

0.009 

0.012 

0.015 

0.017 

89 

1 

0.017 

0.020 

0.023 

0.026 

0.029 

0.032 

0.035 

88 

2 

0.035 

0.038 

0.041 

0.044 

0.047 

0.049 

0.0£2 

87 

3 

0.052 

0.055 

0.058 

0.061 

0.064 

0.067 

0.070 

86 

4 

0.070 

0.073 

0.076 

0.079 

0.082 

0.085 

O.C87 

85 

5 

0.087 

0.090 

0.093 

0.096 

0.099 

0.102 

0.105 

84 

6 

0.105 

0.108 

0.111 

0.114 

0.117    0.120 

0.123 

83 

7 

0.123 

0.126 

0.129 

0.132 

0.135    0.138 

0.141 

82 

8 

0.141 

0.144 

0.146 

0.149 

0.152    0.155    o.l£8 

•81 

9 

0.158 

0.161 

0.164 

0.167 

0.170    0.173   0.176 

80 

10 

0.176 

0.179 

0.182 

0.185 

0.188    0.191 

0.194 

79 

11 

0.194 

0.197 

0.200 

0.203 

0.206    0.210 

0.213 

78 

12 

0.213 

0.216 

0.219 

0.222 

0.225    0.228    0.231 

77 

13 

0.231 

0.234 

0.237 

0.240 

0.243    0.246    0.249 

76 

14 

0.249 

0.252 

0.256 

0.259 

0.262    0.265 

0.268 

75 

15 

0.268 

0.271 

0.274 

0.277 

0.280 

0.284 

0.287 

74 

16 

0.287 

0.290 

0.293 

0.296 

0.299 

0.303 

0.306 

73 

17 

0.306 

0.309 

0.312 

0.315 

0.318 

0.322 

0.325 

72 

18 

0.325 

0.328 

0.331 

0.335 

0.338 

0.341 

0.344 

71 

19 

0.344 

0.348 

0.351 

0.354 

0.357 

0.361 

0.364 

70 

20 

0.364 

0.367 

0.371 

0.374 

0.377 

0.381 

0.384 

69 

21 

0.384 

0.387 

0.391 

0.394 

0.397 

0.401 

0.404 

68 

22 

0.404 

0.407 

0.411 

0.414 

0.418 

0.421 

0.424 

67 

23 

0.424 

0.428 

0.431 

0.435 

0.438 

0.442 

0.445 

66 

24 

0.445 

0.449 

0.452 

0.456 

0.459 

0.463 

0.466 

65 

25 

0.466 

0.470 

0.473 

0.477 

0.481 

0.484 

0.488 

64 

26 

0.488 

0.491 

0.495 

0.499 

0.502 

0.506 

0.510 

63 

27 

0.510 

0.513 

0.517 

0.521 

0.524 

0.528 

0.532 

62 

28 

0.532 

0.535 

0.539 

0.543 

0.547 

0.551 

0.554 

61 

29 

0.554 

0.558 

0.562 

0.566 

0.570 

0.573 

0.577 

60 

30 

0.577 

0.581 

0.585 

0.589 

0.593 

0.597 

0.601 

59 

31 

0.601 

0.605 

0.609 

0.613 

0.617 

0.621 

0.625 

58 

32 

0.625 

0.629 

0.633 

0.637 

0.641 

0.645 

0.649 

57 

33 

0.649 

0.654 

0.658 

0.662 

0.666 

0.670 

0.675 

56 

34 

0.675 

0.679 

0.683 

0.687 

0.692 

0.696 

0.700 

55 

35 

0.700 

0.705 

0.709 

0.713 

0.718 

0.722 

0.727 

54 

36 

0.727 

0.731 

0.735 

0.740 

0.744 

0.749 

0.754 

53 

37 

0.754 

0.758 

0.763 

0.767 

0.772 

0.777 

0.781 

52 

38 

0.781 

0.786 

0.791 

0.795 

0.800 

0.805 

0.810 

51 

39 

0.810 

0.815 

0.  819  !  0.824 

0.829 

0.834 

0.839 

50 

40 

0.839 

0.844 

0.849 

0.854 

0.859    0.864 

0.869 

49 

41 

0.869 

0.874 

0.880 

0.885 

0.890    0.895 

0.900 

48 

42 

0.900 

0.906 

0.911 

0.916 

0.922    0.927 

0.933 

47 

43 

0.933 

0.938 

0.943 

0.949 

0.955    0.960 

0.966 

46 

44 

0.966 

0.971 

0.977 

0.983 

0.988 

0.994 

1.000 

45 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

DEG. 

MINUTES. 

DEG. 

COTANGENTS 


COTANGENTS 


107 


MINUTES. 

DEC. 

DEG. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

0 

oo 

343.8      171.9 

114.6 

85.94 

68.75 

57.29 

89 

1 

57.  29 

49.10 

42.96 

38.19 

34.37 

31.24 

28.64 

88 

2 

28.64 

26.43 

24  .  54 

22.90 

21.47 

20.21 

19.08 

87 

3 

19.08 

18.07 

17.17 

16.35 

15.60 

14.92 

14.30 

86 

4 

14.30 

13.73 

13.20 

12.71 

12.25 

11.83 

11.43 

85 

5 

11.43 

11.06 

10.71 

10.39 

10.08 

9.788 

9.514 

84 

6 

9.514 

9.225 

9.010 

8.777 

8.556 

8.345 

8.144 

83 

7 

8.144 

7.953 

7.770 

7.596 

7.429 

7.269 

7.115 

82 

8 

7.115 

6.968 

6.827 

6.691 

6.561 

6.435 

6.314 

81 

9 

6.314 

6.197 

6.084 

5.976 

5.871 

5.769 

5.671 

80 

10 

5.671 

5.576 

5.485 

5.396 

5.309 

5.226 

5.145 

79 

11 

5.145 

5.066 

4.989 

4.915 

4.843 

4.773 

4.705 

78 

12 

4.705 

4.638 

4.574 

4.511 

4.449 

4.390 

4.331 

77 

13 

4.331 

4.275 

4,219 

4.165 

4.113 

4.061 

4.011 

76 

14 

4.011 

3.962 

3.914 

3.867 

3.821 

3.776 

3.732 

75 

15 

3.732 

3.689 

3.647 

3.606 

3.566 

3.526 

3.487 

74 

16 

3.487 

3.450 

3.412 

3.376 

3.340 

3.305 

3.271 

73 

17 

3.271 

3.237 

3.204 

3.172 

3.140 

3.108 

3.078 

72 

18 

3.078 

3.047 

3.018 

2.989 

2.960 

2.932 

2.904 

71 

19 

2.904 

2.877 

2.850 

2.824 

2.798 

2.773 

2.747 

70 

20 

2.747 

2.723 

2.699 

2.675 

2.651 

2.628 

2.605 

69 

21 

2.605 

2.583 

2.560 

2.539 

2.517 

2.496 

2.475 

68 

22 

2.475 

2.455 

2.434 

2.414 

2.394 

2.375 

2.356 

67 

23 

2.356 

2.337 

2.318 

2.300 

2.282 

2.264 

2.246 

66 

24 

2.246 

2.229 

2.211 

2.194 

2.177 

2.161 

2.145 

65 

25 

2.145 

2.128 

2.112 

2.097 

2.081 

2.066 

2.050 

64 

26 

2.050 

2.035 

2.020 

2.006 

1.991 

1.977 

1.963 

63 

27 

.963 

1.949 

1.935 

1.921 

1.907 

1.894 

1.881 

62 

28 

.881 

1.868 

1.855 

1.842 

1.829 

1.816 

1.804 

61 

29 

.804 

1.792 

1.780 

1.767 

1.756 

1.744 

1.732 

60 

30 

.732 

1.720 

1.709 

1.698 

1.686 

1.675 

1.664 

59 

31 

.664 

1.653 

1.643 

.632 

1.621 

1.611 

1.600 

58 

32 

.600 

1.590 

1.580 

.570 

.560 

1.550 

1.540 

57 

33 

1.540 

1.530 

1.520 

.511 

.501 

1.492 

1.483 

56 

34 

1.483 

1.473 

1.464 

.455 

.446 

1.437 

1.428 

55 

35 

1.428 

1.419 

.411 

.402 

.393 

1.385 

1.376 

54 

36 

1.376 

1.368 

.360 

.351 

.343 

1.335 

1.327 

53 

37 

1.327 

1.319 

.311 

.303 

.295 

1.288 

1.280 

52 

38 

1.280 

1.272 

.265 

.257 

.250 

1.242 

1.235 

51 

39 

.235 

1.228 

1.220 

.213 

.206 

1.199 

1.192 

50 

40 

.192 

1.185 

.178 

.171 

.164 

1.157 

1.150 

49 

41 

.150 

1.144 

1.137 

.130 

.124 

1.117 

1.111 

48 

42 

.111 

1.104 

.098 

.091 

.085 

1.079 

1.072 

47 

43 

.072 

1.066 

.060 

.054 

.048 

1.042 

1.036 

46 

44 

.036 

1.030 

.024 

.018 

.012 

1.006 

1.000 

45 

60' 

50' 

40' 

30' 

20' 

10' 

O' 

DTTP 

T^T1/~l 

-L/rj(j  . 

MINUTES. 

J_JEG. 

TANGENTS 


108         SELF-TAUGHT  MECHANICAL  DRAWING 

gents, "  we  read  "Cotangents/'  and  at  the  bottom 
of  the  table  headed  "  Cotangents, "  we  read  "  Tan- 
gents/' The  object  of  this  will  be  presently  ex- 
plained. The  extreme  left-hand  column,  we  find,  is 
headed  "Deg.,"  and  the  following  seven  columns 
are  headed  0',  10',  20',  30',  40',  50'  and  60',  re- 
spectively, these  columns  indicating  the  minutes. 
At  the  bottom  of  the  pages  the  same  numbers  are 
found  but  reading  from  the  right  to  the  left.  The 
values  of  the  functions  marked  at  the  top  are  read 
in  the  table  opposite  the  degrees  in  the  left-hand 
column  and  under  the  minutes  at  top.  The  values 
of  the  functions  marked  at  the  bottom  are  read 
opposite  the  degrees  in  the  right-hand  column  and 
over  the  minutes  at  the  bottom.  For  example,  the 
sine  of  39  °  40 '  or  sin  39  °  40 ',  as  it  is  written  in 
formulas,  is  thus  found  to  be  0.638,  and  the  sine 
of  64°  10'  is  0.900,  this  latter  value  being  read  off 
in  the  second  table,  reading  it  from  the  bottom  up, 
and  locating  the  number  of  degrees  in  the  right- 
hand  column. 
As  further  examples,  we  find 

tan  37°  40 '  =  0.772 
cot  37°  40 '  =  1.295 
tan80°  0'  =  5.671 
cos  75°  30 '  =  0.250 

We  are  now  ready  to  proceed  to  solve  right-ang- 
led triangles  with  regard  both  to  the  sides  and  the 
angles.  In  any  right-angled  triangle,  if  either  two 
sides,  or  one  side  and  one  of  the  acute  angles  are 
known,  the  remaining  quantities  can  be  found.  As 
a  general  rule,  in  any  triangle,  all  the  quantities 


ELEMENTS  OF  TRIGONOMETRY 


109 


can  be  found  when  three  quantities,  at  least  one 
of  which  is  a  side,  are  given.  In  a  right-angled 
triangle  the  right  angle  is  always  known,  of 
course,  so  that  here,  therefore,  only  two  additional 
quantities  are  necessary.  If  all  the  three  angles 
are  known,  the  length  of  the  sides  cannot  be  de- 
termined; one  side,  at  least,  must  also  always  be 
known  in  order  to  make  possible  the  solution  of 
the  triangle. 

The  following  rules  should  be  used  for  solving 
right-angled  triangles. 

Case  1.  Two  sides  known. — Use  the  rules  al- 
ready given  in  this  chapter  for  finding  the  third 


T 

i 


r ' 1 


GIVEN 
ANGL 


< ADJACENT  SIDE >j 

FIG.  92. 


side  when  two  sides  in  a  right-angled  triangle  are 
given.  To  find  the  angles  use  the  rules  already 
given  for  finding  sines,  cosines,  etc.,  and  the 
tables. 

Case  2.  Hypotenuse  and  one  angle  given. — Call 
the  side  adjacent  to  the  given  angle  the  adjacent 
side,  and  the  side  opposite  the  given  angle  the 
opposite  side  (see  Fig.  92.)  Then  the  adjacent 
side  equals  the  hypotenuse  multiplied  by  the  cosine 


110         SELF-TAUGHT  MECHANICAL  DRAWING 

of  the  given  angle;  the  opposite  side  equals  the 
hypotenuse  multiplied  by  the  sine  of  the  given 
angle ;  and  the  unknown  angle  equals  90  degrees 
minus  the  given  angle. 

Case  3.  One  angle  and  its  adjacent  side  given. 
—The  hypotenuse  equals  the  adjacent  side  divided 
by  the  cosine  of  the  given  angle ;  the  opposite  side 
equals  the  adjacent  side  multiplied  by  the  tangent 
of  the  given  angle;  and  the  unknown  angle  is 
found  as  in  Case  2. 

Case  4.  One  angle  and  its  opposite  side  known. 
—The  hypotenuse  equals  the  opposite  side  divided 
by  the  sine  of  the  given  angle;  the  adjacent  side 
equals  the  opposite  side  multiplied  by  the  cotangent 
of  the  given  angle;  and  the  unknown  angle  is 
found  as  in  Case  2. 

These  rules  may  be  written  as  formulas  as  fol- 
lows (see  Fig.  93) : 

Case  1.  For  formulas  for  the  sides  see  the  first 
part  of  this  Chapter.  For  the  angles  we  have: 

sin  B  =  -  sin  C  =  — . 

a  •     a 

Case  2.    Here,  when  a  and  B  are  given,  we  have : 
c  =  a  cos  B;  b  =  a  sin  B;  C  =  90°  -  B. 

When  a  and  C  are  given,  we  have : 

b  =  a  cos  C;  c  =  a  sin  C;  £  =  90°  -  C. 

Case  3.  Here,  when  B  and  c  are  given,  we  have: 

a  =  — C-^;  b  =  c  tan  B;  C  =  90°  -  B. 
cos  B 

When  C  and  6  are  given,  we  have: 

a  =  — ~;  c  =  6  tan  C;  B  -  90°  -  C. 
cos  C' 


ELEMENTS  OF  TRIGONOMETRY 


111 


Case  4.    Here,  when  B  and  b  are  known,  we 
have: 

,;  c  =  b  cot  B;  C  =  90°  -  B. 


sin 


When  C  and  c  are  known,  we  have  : 


a 


-  ~\  6  -  c  cot  C; 
sin  o 


90°  -  C. 


These  rules  and  formulas,  while  not  including  all 
possible  combinations  for  the  solution  of  right- 
angled  triangles,  give  all  the  information  neces- 
sary for  the  solution  of  any  kind  of  a  right-angled 

A 


C      Bf 


r- 


FIG.  94. 


FIG.  95. 


triangle.  A  few  examples  of  the  use  of  these  rules 
and  formulas  will  now  be  given,  so  as  to  clearly 
indicate  the  mode  of  procedure  in  practical  work. 

Example  1. — In  the  triangle  in  Fig.  94,  side  A  C 
is  12  inches  long  and  angle  D  is  40  degrees.  Find 
angle  E  and  the  two  unknown  sides. 

This  is  an  example  of  Case  3,  one  angle  and  its 
adjacent  side  being  given.  Angle  E  equals  90  de- 
grees minus  the  given  angle,  or 

#=90°  -40°  -50° 


112         SELF-TAUGHT  MECHANICAL  DRAWING 

The  hypotenuse  BC  equals  the  adjacent  side 
divided  by  the  cosine  of  D,  or 

BC  -  -        =  15-666  inches- 


Side  AB  equals  the  adjacent  side  multiplied  by 
the  tangent  of  D,  or 

AB  =  12  X  tan  40°  =  12  X  0.839  =  10.068  inches. 

The  cosine  and  tangent  of  40  degrees  are  found 
in  the  tables  of  trigonometric  functions  as  already 
explained. 

Example  2.  —  In  the  triangle  in  Fig.  95,  the 
hypotenuse  BC  =  17i  inches.  One  angle  is  44  de- 
grees. Find  angle  E  and  the  sides  AB  and  AC. 

This  is  an  example  of  Case  2,  the  hypotenuse 
and  one  'angle  being  given.  Using  the  rules  or 
formulas  given  for  Case  2,  we  have  : 

AC  =  174  X  cos  44°  =  17.5  X  0.719  =  12.5825 

inches. 
AB  =  174  X  sin  44°  =  17.5  X  0.695  =  12.1625 

inches. 

E  =90°  -44°  =46°. 

Example  3.  —  In  the  triangle  in  Fig.  96,  side  AC 
=  208  feet,  and  the  angle  opposite  this  side  =  38 
degrees.  Find  angle  E,  and  the  two  remaining 
sides. 

This  is  an  example  of  Case  4,  one  side  and  the 
angle  opposite  it  being  known.     From  the  rules  or 
formulas  given  for  Case  4,  we  have: 
BC  =  208  -  sin  38°  =  208  -  0.616  =  337.66  feet. 
AB  =  208  X  cot  38°  =  208  X  1.280  =  266.24  feet. 
#=90°  -38°  =  52°. 


ELEMENTS  OF  TRIGONOMETRY 


113 


Example  4. — In  the  triangle  in  Fig.  97,  side  AC 
=  3  inches,  and  the  hypotenuse  #C=5  inches.  Find 
side  AB  and  angles  D  and  E. 

This  is  an  example  of  Case  1.  According  to  a 
formula  previously  given  in  this  chapter 


AB  =  VBC'2-  AC2  = 

Vl6  =  4. 

AB 


sin  E  = 


BC 


\/52-  32  =  \/25  -  9  = 


=0.800. 


From  the  tables  we  find  that  the  angle  corre- 


FIG.  96. 


FIG.  97. 


spending  to  a  sine  which  equals  0.800  is  53°  10'. 
Consequently : 
#  =  53°  10',  and  D  =  90°  -53°  10' =  36°  50'. 

Example  5. — In  the  triangle  in  Fig.  98,  side  BC, 
the  hypotenuse,  is  1|  inch  long.  One  angle  is  65 
degrees.  Find  angle  E  and  the  remaining  sides. 


114         SELF-TAUGHT  MECHANICAL  DRAWING 

This  is  an  example  of  Case  2.    We  have: 
E=  90°  -65°  =25°. 

AB=  1|  X  cos  65°  =  1.375  X  0.423  -  0.5816  inch. 
AC  =  1|  X  sin  65°  =  1.375  X  0.906  =1.2457  inch. 

Example  6.— In  the  triangle  in  Fig.  99,  side  AB 
=  0.706  inch,  and  the  angle  adjacent  to  this  side  is 
60  degrees.  Find  angle  E and  the  sides  AC  and  EC. 

A  B 

T 


U ia£ 


FIG.  99. 

This  is  an  example  of  Case  3.     We  have : 

#=90°  -60°  =  30°. 

EC  =  0.706  •*-  cos  60°  -  0.706  -  0.500  =  1.412 

inch. 
AC=  0.706  X  tan  60°  =  0.706  X  1.732=  1.2228 

inch. 

The  previous  examples,  carefully  studied,  will 
give  a  comprehensive  idea  of  the  methods  used  for 
solving  right-angled  triangles,  no  matter  which 
parts  are  given  or  unknown. 

A  triangle  which  does  not  contain  a  right  angle 
is  called  an  oblique  triangle.  Any  such  triangle 
can  be  solved  by  the  aid  of  the  formulas  given  for 
the  right  triangle,  by  dividing  it  into  two  right- 
angled  triangles  by  means  of  a  line  drawn  from 
the  vertex  of  one  angle  perpendicular  towards  the 
opposite  side.  Formulas  can  be  deduced  which  do 


ELEMENTS  OF  TRIGONOMETRY 


115 


not  require  that  the  triangle  be  so  divided,  but  for 
elementary  purposes,  the  method  indicated  is  the 
most  easily  understood. 

In  Fig.  100,  for  example,  a  triangle  is  given  as 
shown.  One  angle  is  50  degrees,  and  the  sides  in- 
cluding this  angle  are  4  and  5  inches  long,  respec- 
tively. Draw  a  line  from  A  perpendicular  to  the 


side  EC.  We  have  here  two  right-angled  tri- 
angles, and  can  now  proceed  by  using  the  formulas 
previously  given.  In  triangle  ADB,  the  hypoten- 
use AB  and  one  angle  are  given.  We  then  find 
side  AD  by  means  of  the  formulas  for  Case  2,  and 
also  angle  BAD  and  side  BD.  Next  we  find  CD 
=  5-  BD.  We  then,  in  the  triangle  A  CD  know 
two  sides  AD  and  CD,  and  can  thus  find  side  AC 
as  in  Case  1,  as  well  as  angles  A  CD  and  CAD. 
The  angle  BAG  finally  is  found  by  adding  angles 


116         SELF-TAUGHT  MECHANICAL  DRAWING 

BAD  and  CAD  and,  then,  all  the  angles  and  sides 
in  the  triangle  are  found. 

The  successive  calculations  would  be  carried  out 
as  follows: 

AD  =  4  X  sin  50°  =  4  X  0.766  -  3.064. 

£D  =  4  X  cos  50°=  4  X  0.643  =  2.572. 

Angle  BAD  =  90°  -  50°  =  40°. 

DC  =  5  -  BD  =  5  -  2.572  =  2.428. 


AC  =    AD*  +  DC*  =    aoe      2.428*  =-  3.91. 


Sine  of  angle  ACD  =  ~  =  ~       -  0.784. 


Angle  A  CD  =  51°  40'. 

Angle  CAD  =  90°-  51°  40'  =  38°  20'. 
Angle  BAG  =  40°  +  38°  20'=  78°  20'. 

In  order  to  check  the  results  obtained,  add  angles 
ABC,  BAG  and  ACD.  The  sum  of  these  angles 
must  equal  180  degrees  if  the  results  are  correct: 

50°  +  78°  20'  +  51°  40'  =  180°. 

This  method,  with  such  modifications  as  are 
necessary  to  meet  the  different  requirements  in 
each  problem,  may  be  used  for  solving  all  oblique- 
angled  triangles,  except  in  the  case  where  no  angle 
is  known,  but  only  the  lengths  of  all  the  three 
sides.  In  this  case  the  use  of  a  direct  formula 
will  prove  the  best  and  most  convenient.  Let  the 
three  known  sides  be  a,  b  and  c,  and  the  angles 
opposite  each  of  them  A,  B  and  C,  respectively,  as 
in  Fig.  101  ;  then  we  have  : 

b2  +  c2  -  a2  b  sin  A 


180°-  (A  +  B). 


ELEMENTS  OF  TRIGONOMETRY 


117 


As  an  example,  assume  that  the  three  sides  in  a 
triangle  are  a  =  4,  6  =  5,  and  c  =  6  inches  long. 
Find  the  angles. 

52  +  62-  42  =  45_ 
60 


Cos  4  = 


2X5X6 
,4  =  41°  25'. 


0.750. 


Sin  B  = 


_ 


4  4 

B  =  55°  50'. 
C  =  180°  -  (41°  25'  +  55°  50')  =  82°  45'. 

As  only  the  first  principles  of  trigonometry  have 
here  been  treated,  some  of  the  more  advanced 


^ 


problems  have,  by  necessity,  been  omitted.  For 
ordinary  shop  calculations  the  present  treatment 
will,  however,  be  found  more  satisfactory,  as  some 
of  the  matter  which  would  unnecessarily  burden 
the  mind  has  been  left  out.  If  the  student  only 
first  acquires  a  thorough  understanding  of  the  first 


118         SELF-TAUGHT  MECHANICAL  DRAWING 

principles  of  mathematics  and  their  application  to 
machine  design,  it  is  comparatively  easy  to  broaden 
the  field  of  one's  knowledge;  it  is,  therefore,  of 
extreme  importance  that  these  first  principles 
be  thoroughly  understood  and  digested.  The  ap- 
plication will  then  be  found  comparatively  easy. 

The  trigonometric  functions  afford  a  convenient 
means  for  laying  out  angles ;  and  when  the  sides 


Ar  — 

r 60 >j 


^FiG.  102.— Method  of  Laying  Out  Angles  by  Means  of 
Natural  Functions. 


of  the  angle  laid  out  are  much  extended,  it  can 
be  laid  out  more  accurately  in  this  manner  than 
by  the  use  of  an  ordinary  protractor.  Let  it  be 
required,  for  instance,  to  lay  out  an  angle  of  37 
degrees,  one  side  of  the  angle  being  60  inches  long. 
Lay  out  the  side  AB,  Fig.  102,  60  inches  long. 
Then  with  a  radius  equal  to  the  sine  of  37  degrees 
multiplied  by  60,  and  with  a  center  at  B,  draw  an 


ELEMENTS  OF  TRIGONOMETRY 


119 


arc  C.  Then  draw  a  line  from  A,  tangent  to  arc 
C.  This  line  forms  an  angle  of  37  degrees  with 
line  AB.  If  the  required  angle  is  over  45  degrees, 
then  it  is  preferable  to  lay  out  the  complement 
angle  from  a  line  perpendicular  to  the  original 


-B 


FIG-  103.— Laying  Out  an  Angle  Greater  than  45  Degrees. 

line,  as  shown  in  Fig.  103,  where  an  angle  of  70 
degrees  is  to  be  laid  out,  but  the  20-degree  comple- 
ment angle  is  actually  constructed.  Many  other 
methods  for  use  in  laying  out  angles,  arcs,  etc., 
will  readily  suggest  themselves  to  the  student  who 
thoroughly  understands  the  relation  of  the  trigo- 
nometric functions  in  a  right-angled  triangle. 


CHAPTER  VIII 

ELEMENTS  OF  MECHANICS 

MECHANICS  is  defined  as  that  science,  or  branch 
of  applied  mathematics,  which  treats  of  the  action 
of  forces  on  bodies.  That  part  of  mechanics  which 
considers  the  action  of  forces  in  producing  rest  or 
equilibrium  is  called  statics;  that  which  relates  to 
such  action  in  producing  motion  is  called  dynamics; 
the  term  mechanics  includes  the  action  of  forces 
on  all  bodies  whether  solid,  liquid  or  gaseous.  It 
is  sometimes,  however,  and  formerly  was  often, 
used  distinctively  of  solid  bodies  only.  The  me- 
chanics of  liquid  bodies  is  called  also  hydrostatics 
or  hydrodynamics,  according  as  the  laws  of  rest  or 
motion  are  considered.  The  mechanics  of  gaseous 
bodies  is  called  also  pneumatics.  The  mechanics 
of  fluids  in  motion,  with  special  reference  to  the 
methods  of  obtaining  from  them  useful  results, 
constitutes  hydraulics. 

The  Resultant  of  Two  or  More  Forces.— When  a 
body  is  acted  upon  by  several  forces  of  different 
magnitudes  in  different  directions,  a  single  force 
may  be  found,  which  in  direction  and  magnitude 
will  be  a  resultant  of  the  action  of  the  several 
forces.  The  magnitude  and  direction  of  this  single 
force  may  be  obtained  by  what  is  known  as  the 
parallelogram  of  forces.  Let  A  and  B,  Fig.  104, 

120 


ELEMENTS  OF  MECHANICS  121 

represent  the  direction  of  two  forces  acting  simul- 
taneously upon  P,  and  let  their  lengths  represent 
the  relative  magnitude  of  the  forces ;  then,  to  find 
a  force  which  in  direction  and  magnitude  shall 
be  a  resultant  of  these  two  forces,  draw  the  line  C 
parallel  with  B,  and  draw  the  line  D  parallel  with 
A.  A  diagonal  of  the  parallelogram  thus  formed, 
drawn  from  Pto  E,  will  give  the  direction,  and  its 


F 

FIG.  104.— Parallelogram  of  Forces. 

length  as  compared  with  A  and  B,  the  relative 
magnitude,  of  the  required  force. 

That  this  is  so  may  be  seen  by  considering  the 
two  forces  as  acting  separately  upon  P.  Let  A  be 
considered  as  acting  upon  P  to  move  it  through 
a  distance  equal  to  its  length.  Then  P  would  be 
moved  to  F.  If  the  force  B  is  now  caused  to  act 
upon  P  to  move  it  through  a  distance  equal  to  its 
length,  P  will  arrive  at  G.  As  FP  has  the  same 
length  and  direction  as  A,  and  as  GFhas  the  same 
length  and  direction  as  B,  the  distance  from  G  to 
P  would  be  the  same  as  the  distance  from  P  to  E; 
therefore,  PE,  the  diagonal  of  the  parallelogram 
formed  by  the  lines  A,  B,  C,  and  D,  represents  the 
required  new  force  or  resultant. 

If  there  are  more  than  two  forces  acting  upon 
the  point  P,  first  find  a  resultant  of  any  two  of  the 
forces;  then  consider  this  resultant  as  replacing 


122 


SELF-TAUGHT  MECHANICAL  DRAWING 


FIG.  105.  -Resultant  of  Three 
Forces. 


the  first  two,  and  find  the  resultant  of  it  and  an- 
other of  the  original  forces ;  continue  this  process 
until  a  force  is  obtained  which  will  be  the  resultant 
of  all  of  the  original  forces.  Thus,  in  Fig.  105,  if 
A,  B  and  C  be  considered  as  representing  in  di- 
rection and  magnitude 
three  forces  which  are 
acting  simultaneously 
uponP;  then,  if  we  draw 
a  parallelogram  upon  A 
and  B,  we  have  its  diag- 
onal PD  as  the  resultant 
of  A  and  B.  A  parallel- 
ogram is  now  drawn 
upon  PD  and  C,  giving  PE,  its  diagonal,  as  the 
resultant  of  these  two,  and,  consequently,  of  the 
three  original  forces. 

This  principle  holds  true  whether  the  original 
forces  are  acting  in  the  same  plane  or  not.  Thus, 
in  Fig.  106,  let  A,  B 
and  C  be  three  forces 
acting  simultaneously 
upon  P.  Then  the  re- 
sultant of  A  and  B 
would  be  the  diagonal 
PD.  Considering  this 
as  replacing  A  and  J5, 
a  resultant  of  it  and  C 

would  be  a  diagonal  drawn  from  P  to  the  further 
corner  E;  PE  would  then  be  the  resultant  of  A,  B 
andC. 

This  operation  may,  of  course,  be   reversed  to 
allow  of  finding  two  or  more  forces  in  different 


FIG.  106.  -Resultant  of  Three 
Forces  in  Different  Planes. 


ELEMENTS  OF  MECHANICS 


123 


directions  which  in  magnitude  shall  be  equivalent 
to  a  single  known  force.  Thus  in  Fig.  107,  if  PA 
represents  the  direction  and  magnitude  of  a  given 
force  which  it  is  desired  to  replace  by  two  others 
acting  in  the  direction  of  PB  and  PC,  respectively, 
then  draw  a  line  from  A  to  PB  parallel  with  PC, 
and  draw  another  from  A  to  PC  parallel  with  PB. 
The  lengths  Pa  and  Pb 
thus  determined  will 
represent  the  relative 
magnitudes,  as  com- 
pared with  PA,  of  the 
required  new  forces. 

Parallel  Forces.— Let 
A  and  B,  Fig.  108, 
represent  the  direction 
and  magnitude  of  two 
parallel  forces  acting 

together  upon  the  bar  DE.  These  two  forces  may 
be  replaced  or  counterbalanced  by  a  single  force, 
equal  in  magnitude  to  A  and  B  combined.  To  de- 
termine the  point  of  application  of  this  new  force 
produce  A  to  a,  making  Da  equal  in  length  to  B. 
Also  make  bE  equal  in  length  to  A.  The  inter- 
section of  the  line  connecting  a  and  b  with  DE,  at 
F,  will  be  the  required  point  of  application.  The 
lengths  DF  and  FE  will  be  inversely  proportional 
to  the  forces  A  and  B.  That  is,  the  length  FE  will 
be  to  the  force  A  as  the  length  DF  is  to  the  force 
B.  The  product  of  DF  multiplied  by  A  will  be 
equal  to  the  product  of  FE  multiplied  by  B. 

Fig.  109  shows  how  several  parallel  forces,  act- 
ing in  the   same  direction,  may  be  replaced  or 


FIG.  107.— Resolution  of  Forces. 


124         SELF-TAUGHT  MECHANICAL  DRAWING 


counterbalanced  by  a  single  force.  Let  A,  B  and 
C  represent  the  relative  magnitudes  of  the  forces. 
A  resultant  of  B  and  C  would  be  D,  equal  in 


/ 

xi. 


FIG.  108.— Parallel  Forces. 


FIG.  109.— Resultant  of 
Several  Parallel  Forces. 


magnitude  to  B  and  C  combined,  and  its  point  of 
application,  determined  in  the  manner  previously 
described,  would  be  at  a.  Regarding  D  as  a  single 

force  replacing  B  and  C, 
would  give  E,  equal  in 
magnitude  to  A  and  D 
combined,  as  the  result- 
ant of  these  two,  and  its 
point  of  application,  de- 
termined as  before,  would 
be  at  b. 

Oblique  Forces. — Let  A 
and  B,  Fig.  110,  repre- 
sent the  directions  and 
relative  magnitudes  of 

two  forces  acting  simultaneously  upon  the  barDE. 
These  two  forces  may  be  either  replaced  or  counter- 


FIG.  110.— Oblique-  Forces 
Acting  at  Different  Points 
on  a  Bar. 


ELEMENTS  OF  MECHANICS 


125 


balanced  by  a  single  force,  which  in  direction  and 
magnitude  shall  be  a  resultant  of  them.  Produce 
A  and  B  until  they  meet  at  a.  Draw  the  parallel- 
ogram abed,  making  da  equal  to  A,  and  ba  equal 
to  B.  The  diagonal  of  this  parallelogram  will  give 
the  direction  and  relative  magnitude  of  the  new 
force,  and  if  extended  its  intersection  with  DE 
will  give  the  point  of  application. 

Opposing  Forces. — Let  A  and  J9,  Fig.  Ill,  repre- 
sent the  directions  and  relative  magnitudes  of  two 
forces  acting  upon  oppo- 
site sides  of  the  bar  DE. 
These  two  forces  may  be 
replaced  by  a  single  force, 
which  in  direction  and 
magnitude  will  be  a  re- 
sultant of  them.  Produce 
A  and  B  until  they  meet 
at  a.  Lay  off  ac  equal  to 
the  length  of  B,  and  make 
be  equal  to  and  parallel 
with  A.  A  line  drawn 
from  a  to  6  will  give  the 

direction  of  the  new  force,  and  the  length  of  ab, 
as  compared  with  A  and  B  will  give  its  relative 
magnitude.  Its  application  on  bar  DE  may  be  de- 
termined by  extending  ab  until  it  intersects  DE. 

Levers. — When  a  workman  wishes  to  raise  a 
heavy  object,  he  may  insert  one  end  of  a  bar  un- 
der it,  and  lift  on  the  other  end;  or,  pushing  a 
block  of  wood  or  iron  in  under  the  bar  as  close  to 
the  object  to  be  raised  as  he  can,  he  presses  down 
upon  the  free  end  of  the  bar.  A  bar  so  used  con- 


FIG.  111. — Opposing  Oblique 
Forces. 


126         SELF-TAUGHT  MECHANICAL  DRAWING 

stitutes  a  lever,  and  the  point  where  the  bar  rests 
when  the  lever  is  doing  its  work,  the  end  of  the 
bar  in  under  the  heavy  object  in  the  first  case,  or 
the  block  on  which  the  bar  rests  in  the  second 
case,  is  the  fulcrum  of  the  lever. 

Levers  are  of  three  kinds,  as  shown  in  Fig.  112: 
First,  where  the  fulcrum  is  between  the  power 


J1ST 


I  3RD 

Ow 


FIG.  112.— Classes  of  Levers. 

and  the  weight;  second,  where  the  weight  is 
between  the  fulcrum  and  the  power;  and,  third, 
where  the  power  is  between  the  fulcrum  and  the 
weight.  A  man's  forearm  furnishes  a  good  illus- 
tration of  a  lever  of  the  third  class,  the  fulcrum 
being  at  the  elbow,  the  weight  at  the  hand,  and 
the  muscle,  being  attached  to  the  bone  of  the  arm, 
at  a  short  distance  from  the  elbow,  furnishing  the 
power. 


ELEMENTS  OF  MECHANICS  127 

In  all  of  these  cases  the  gain  in  power  is  exactly 
proportional  to  the  loss  in  speed,  or  the  gain  in 
speed  is  exactly  proportional  to  the  loss  in  power. 
Also,  in  every  case  the  product  of  the  weight  mul- 
tiplied by  its  distance  from  the  fulcrum,  will  equal 
the  product  of  the  power  multiplied  by  its  distance 
from  the  fulcrum,  or,  the  weight  and  power  will 
balance  each  other  when  the  weight  multiplied  by 
the  distance  through  which  it  moves,  equals  the 
power  multiplied  by  the  distance  through  which  it 
moves. 

If  in  Fig.  108  the  bar  DE  is  a  lever,  the  fulcrum 
will  be  at  F,  and  the  methods  used  in  that  figure 
and  in  Figs.  109,  110  and  111  give  solutions  of  dif- 
ferent lever  problems. 

The  length  of  the  lever  arm  is  independent  of 
the  form  of  the  lever.  In  Fig.  113  is  shown  a  lever 


FIG.  113. -Lever  of  Curved  Shape. 

of  curved  shape ;  but  the  lever  arms  on  which  the 
calculation  as  to  the  work  that  the  lever  is  doing, 
will  be  based,  will  be  straight  lines  connecting  the 
point  where  the  power  is  applied,  or  the  point 
which  supports  the  weight,  with  the  fulcrum. 

The  length  of  the  lever  arm  is  always  at  right 
angles  to  the  direction  in  which  the  power  is  being 


128 


SELF-TAUGHT  MECHANICAL  DRAWING 


applied,  or  to  the  direction  of  the  resistance  of  the 
weight  or  load. 

In  Fig.  114  two  cases  are  shown  where  the  power 
is  applied  obliquely  on  the  lever;  but  the  lever  arm 
on  which  the  calculation  is  based  will  be  the  dis- 


f\ 


FIG.  114.— Power  Applied  Obliquely  on  Lever. 

tance  Fa  measured  from  the  fulcrum,  at  right 
angles  to  the  direction  of  the  power. 

Compound  Levers. — In  Fig.  115  is  shown  a  case 
where  the  power  gained  with  one  lever  is  further 
increased  by'  the  use  of  a  second  lever,  acting  on 
the  first  one.  The  weight  and  power  will  balance 


ELEMENTS  OF  MECHANICS  129 

each  other  when  the  product  of  the  weight  and  the 
lever  arms  ab  and  ef,  multiplied  together,  equals 
the  product  of  the  power  and  the  lever  arms  gfand 
be  multiplied  together.  Thus,  to  find  the  weight 


1                                                             1 

(_> 

^F 

9 

V 

I                                                                                        I 

1                                                                         e           > 

FIG.  115.— Compound  Levers. 

which  a  given  power  will  lift,  divide  the  product 
of  the  power  and  its  lever  arms  &/*and  be,  multi- 
plied together,  by  the  product  of  the  lever  arms  of 
the  weight,  ab  and  ef,  multiplied  together.  To  find 


6W 


A  B 

FIG.  116.— Diagram  for  Lever  Problem. 

the  power  necessary  to  lift  a  given  weight,  divide 
the  product  of  the  weight  and  its  lever  arms,  ab 
and  ef,  multiplied  together,  by  the  product  of  the 
lever  arms  of  the  power,  gf  and  be,  multiplied 
together. 


130         SELF-TAUGHT  MECHANICAL  DRAWING 

A  few  examples  will  illustrate  these  principles. 
Assume  that  in  Fig.  116  a  weight  at  A  must  bal- 
ance the  18-pound  weight  at  B.  The  lever  arms 
are  given  as  12  and  5  inches,  respectively.  How 
much  must  the  weight  W  be,  in  order  to  balance 
the  weight  at  B  ? 

The  weight  at  B  (18  pounds)  times  its  lever  arm 
(5  inches)  must  equal  the  weight  W  times  its  lever 
arm  (12  inches).  In  other  words: 

18  X  5  =  W  X  12. 
90  =  12  W. 

W  =  12  =  7i  P°unds- 

In  Fig.  117,  two  weights,  4  and  2  pounds,  respec- 
tively, are  balanced  by  a  weight  W.  Find  what 


r 


./ 

c 


FIG.  117.  —  Diagram  for  Lever  Problem. 


the  weight  of  TFmust   be  with  the  lever   arms 
given  in  the  engraving. 

In  this  case  the  weight  at  A  times  its  lever  arm 
plus  the  weight  at  B  times  its  lever  arm,  will 
equal  weight  W  times  its  lever  arm.  The  sum  of 
the  products  of  the  weights  and  leverages  of  the 
weight  at  A  and  B  is  taken,  because  both  these 
weights  are  on  the  same  side  of  the  fulcrum  F. 


ELEMENTS  OF  MECHANICS  131 

Carrying  out  the  calculation   outlined  above,  we 

have: 

4  X  16  +  2X8  =  6  W. 
64  +  16  =  80  =  6  W. 

W  =  8^-  =  134  pounds. 
b 

The  product  of  a  weight  or  force  and  its  lever 
arm  is  commonly  called  the  moment  of  the  force. 
The  moment  of  the  force  at  A,  for  example,  is  4 
pounds  X 16  inches  =  64  inch-pounds.  If  the  lever 
arm  were  16  feet  instead  of  16  inches,  the  result 
would  be  64  foot-pounds. 

An  interesting  application  of  the  lever,  and  the 
moments  of  forces,  is  presented  in  calculations  of 


FlG.  118. — Diagram  for  Lever  Problem. 

weights  for  safety  valves.  A  diagrammatical 
sketch  of  a  safety  valve  lever  is  shown  in  Fig.  118. 
Assume  that  the  total  steam  pressure,  acting  on 
the  whole  area  of  the  safety  valve,  is  300  pounds 
when  it  is  required  that  the  steam  should  "blow 
off."  Find  the  weight  W  required  near  the  end 
of  the  lever  to  keep  the  valve  down  until  the  total 
pressure  is  300  pounds  on  the  valve.  Assume  the 
weight  of  the  lever  itself  to  be  6  pounds,  con- 
centrated at  its  center  of  gravity,  10  inches  from 
the  fulcrum  F. 


132        SELF-TAUGHT  MECHANICAL  DRAWING 

In  this  case  we  have  that  the  moment  of  the 
steam  pressure,  which  acts  upward,  should  equal 
the  sum  of  the  moments  of  the  weight  of  the  lever 
and  the  weight  W.  Therefore  : 

300  X  3  =  6  X  10  +  20  W. 
900  =  60  +  20  W. 
900  -  60  =  20  W. 
840  =  20  W. 

TJ7      840        ,0          , 
W  =  -gQ-  =  42  pounds. 

The  calculation  above  has  been  carried  out  step 
by  step,  so  that  students  unfamiliar  with  the  alge- 
braic solution  of  equations  may  be  able  to  under- 
stand the  principles  involved  in  simple  examples 
of  this  kind.  In  the  following,  the  calculations 
have  been  carried  out  more  directly,  but  the  stu- 
dent should  use  the  "step  by  step"  method  until 
thoroughly  familiar  with  the  subject. 

Fixed  and  Movable  Pulleys. — A  fixed  pulley  is 
frequently  used  to  change  the  direction  of  the 
power,  as  shown  in  Fig.  119,  but  there  is  no  gain 
in  power  with  such  a  pulley,  as  there  is  no  com- 
pensating loss  of  speed ;  the  weight  will  move  up- 
ward at  the  same  rate  of  speed  as  the  power  moves 
downward. 

If  now  a  movable  pulley  be  used  in  connection 
with  the  fixed  pulley  as  shown  in  Fig.  120,  then  as 
the  end  of  the  rope  to  which  the  power  is  applied 
is  drawn  downward,  each  of  the  two  strands  of 
rope  between  the  pulleys  will  take  half  of  the 
stress  of  the  suspended  weight,  and  the  weight 
will  be  raised  only  one-half  the  distance  that  the 


ELEMENTS  OF  MECHANICS 


133 


power  descends.  The  power  will  therefore  need  to 
be  only  one-half  of  the  weight.  In  Fig.  121,  there 
are  three  strands  of  rope  between  the  pulleys,  each 
of  which  will  be  equally  shortened  when  the  free 
end  of  the  rope  is  pulled ;  the  power,  therefore,  is 
only  one-third  of  the  weight.  In  Fig.  122,  with 


o 


FIG.  119.— Fixed  Pulley. 


FIG.  120.— Fixed  and  Movable 
Pulleys. 


four  strands  of  rope  between  the  pulleys,  each  fur- 
nishing an  equal  amount  to  the  free  end  as  it  is 
drawn  out,  the  power  need  be  only  one-fourth  of 
the  weight. 

The  law  of  the  pulley,  then,  where  a  single  rope 
is  employed,  is  that  the  power  will  be  increased  as 
many  times  as  there  are  lines  of  rope  between  the 
pulleys  to  participate  in  the  shortening.  In  a  sys- 
tem using  more  than  one  rope,  as  shown  in  Fig. 


134         SELF-TAUGHT  MECHANICAL  DRAWING 


123,  each  additional  movable  pulley  doubles  the 
power,  as  it  will  move  at  only  half  the  rate  of  the 
preceding  pulley. 

Differential  Pulleys. — Another  form  of  pulley, 
known  as  the  differential  pulley,  much  used  in  ma- 
chine shops,  is  shown  in  Fig.  124.  In  this  form  of 


w 


FIG.  121.— Tackle  where  Load 
is  Taken  on  Three  Strands 
of  Rope. 


FIG.  122.— Tackle  where  Load 
is  Taken  on  Four  Strands 
of  Rope. 


pulley  an  endless  chain  replaces  the  rope,  the  pul- 
leys themselves  being  grooved  and  toothed  like 
sprocket  wheels.  The  two  pulleys  at  the  top  are 
of  slightly  different  diameters,  but  rotate  together 
as  one  piece.  In  operation,  as  the  chain  is  drawn 
up  by  the  large  wheel  it  passes  around  in  a  loop  to 
the  small  wheel  from  which  it  is  unwound,  causing 
the  loop  in  which  the  movable  pulley  rests  to  be 


ELEMENTS  OF  MECHANICS 


135 


shortened  by  an  amount  equal  to  the  difference  in 
the  pitch  circumferences  of  the  two  upper  wheels, 
when  they  have  made  one  revolution.  This  would 
cause  the  weight  to  be  raised  one-half  of  that 
amount.  If  in  a  given  case  the  two  upper  pulleys 
had  respectively  20  and  19  teeth,  then  as  the  ap- 


FlG.  123.— A  Special  Arrange- 
ment of  Movable  Pulleys. 


FIG.  124.— Differential 
Pulley. 


plied  power  was  being  moved  through  a  distance 
of  20  inches  the  small  pulley  would  unwind  19 
inches  of  the  chain,  causing  a  shortening  of  the 
loop  in  which  the  movable  pulley  rests  of  one  inch, 
which  would  raise  the  weight  one-half  of  an  inch, 
giving  a  ratio  of  load  to  power  of  40  to  1. 

In  all  of  these  cases  the  results  actually  attained 
in  practice  will  be  somewhat  modified  from  the 


136        SELF-TAUGHT  MECHANICAL  DRAWING 

theoretical  results  given   by  calculations,  by  the 
losses  occasioned  by  friction. 

Inclined  Planes. — In  raising  heavy  weights 
through  short  distances,  as  for  instance  in  loading 
barrels  onto  wagons,  a  plank  may  be  used  to  facili- 
tate the  work  by  placing  one  end  of  it  on  the 
ground  and  the  other  end  on  the  wagon,  and  roll- 
ing the  barrel  up  the  plank  onto  the  wagon.  Such 
an  arrangement  is  called  an  inclined  plane.  When 
the  force  which  is  being  applied  to  the  rolling 


FIG.  125.— Inclined  Plane.  FIG.  126.— Power  Applied 

Parallel  to  Base. 

object  is  exerted  in  a  direction  parallel  to  the  in- 
clined surface,  as  in  Fig.  125,  it  is  evident  that  the 
power  must  move  through  a  distance  equal  to  the 
length  of  the  incline  in  order  to  raise  the  weight 
the  desired  height.  The  gain  in  power  will  then 
be  equal  to  the  length  of  the  incline  divided  by  the 
height. 

If  the  power  is  applied  in  a  direction  parallel 
with  the  base,  as  in  Fig.  126,  the  power  will  have 
to  advance  through  a  distance  equal  to  the  length 
of  the  base  to  raise  the  object  the  desired  height. 
The  gain  in  power  will  then  be  equal  to  the  base 
divided  by  the  height.  By  considering  Fig.  126 


ELEMENTS  OF  MECHANICS 


137 


FIG.  127.— Power  Applied  Obliquely 
to  Surface  of  Incline. 


further,  it  will  be  seen  that  in  rolling  the  object 
up  the  incline  the  power  will  have  to  advance  from 
the  beginning  of  the 
incline  to  a  point 
from  which  a  line 

may  be  drawn  per-  ' 

pendicular  to  its  di- 
rection to  the  top  of 
the  incline.  In  any 
case  where  the 
power  is  applied  in 
any  direction  other 
than  parallel  with 
the  incline,  in  roll- 
ing the  object  to  the  top,  the  power  will  have  to 
advance  to  a  point  from  which  a  line  may  be  drawn 
perpendicularly  to  its  direction  to  the  top  of  the 

incline.  In  Figs.  127 
and  128  are  shown  two 
other  cases  where  the 
power  is  applied  in 
a  direction  obliquely 
to  the  surface  of  the 
incline.  In  either  of 
these  cases,  as  in  the 
other  two  cases,  the 
gain  in  power  will 
be  found  by  dividing 
the  distance  through 
which  the  force 
distance  through  which  the 


FIG.  128.— Another  Case  where 
Power  is  Applied  Obliquely  to 
Surface  of  Incline. 


moves,  ab,   by  the 
object  is  raised,  cd. 
It  will  be  further  seen  that  the  gain  in  power  is 


138         SELF-TAUGHT  MECHANICAL  DRAWING 


greatest  when  the  direction  in  which  the  force  is 
being  applied  is  parallel  with  the  incline.  When 
the  direction  of  the  force  is  upward  from  the  in- 
cline, as  in  Fig.  127,  part  of  the  force  is  expended 
in  lifting  the  weight  off  from  the  incline,  until, 
when  its  direction  is  made  vertical,  it  is  all 
expended  in  this  way.  When  the  direction  of  the 
force  is  downward  from  the  incline,  as  in  Figs.  126 
and  128,  part  of  it  is  lost  in  pressing  the  object 
against  the  incline. 

The  Screw. — The  screw  is  a  modified  form  of 
inclined  plane,  the  lead  of  the  screw,  the  distance 


FIG.  129.— Differential  Screw. 

that  the  thread  advances  in  going  around  the 
screw  once,  being  the  height  of  the;  incline,  and 
the  distance  around  the  screw,  measured  on  the 
thread,  being  the  length  of  the  incline. 

The  Differential  Screw.— The  differential  screw 
is  a  compound  screw  having  a  coarse  thread  part  of 
its  length,  and  a  somewhat  finer  thread  the  rest  of 
its  length,  the  object  being  to  get  a  slow  motion 
combined  with  the  strength  of  a  coarse  thread. 
Fig.  129  shows  such  a  screw.  The  piece  A  is  a 
fixed  part  of  some  machine.  The  piston  B  slides 
within  A,  being  prevented  from  turning  by  the  pin 
C  which  enters  a  groove  in  B.  If  that  part  of  the 


ELEMENTS  OF  MECHANICS  139 

screw  which  engages  in  A  has  eight  threads  to 
the  inch,  and  that  part  of  it  which  engages  in  B 
has  ten  threads,  then  when  the  screw  makes  one 
revolution,  it  will  advance  into  A  one-eighth  of  an 
inch,  and  into  B  one- tenth  of  an  inch ;  the  piston 
B  will  therefore  advance  through  a  distance  equal 
to  the  difference  between  one-eighth  of  an  inch 
and  one-tenth  of  an  inch,  or  twenty-five  one-thou- 
sandths of  an  inch,  requiring  forty  turns  of  the 
screw  to  make  the  piston  advance  one  inch. 

Newton's  Laws  of  Motion. — The  relation  which 
exists  between  force  and  motion  is  stated  by  the 
three  fundamental  laws  of  motion  formulated  by 
Newton. 

Newton's  first  law  says  that  if  a  body  is  at 
rest  it  will  remain  at  rest,  or,  if  it  is  in  motion,  it 
will  continue  to  move  at  a  uniform  velocity  in  a 
straight  line,  until  acted  upon  by  some  force  and 
compelled  to  change  its  state  of  rest  or  of  straight- 
line  uniform  motion.  In  a  general  way,  this  law  is 
self-evident,  and  based  on  daily  experience.  How- 
ever, the  part  of  the  law  stating  that  a  body  in 
motion  will  continue  indefinitely  to  move  if  not 
acted  upon  by  resisting  forces,  may  not  be  so  self- 
evident;  yet  whenever  a  body  is  brought  to  a  stand- 
still after  it  has  been  in  motion,  such  forces  as 
frictional  resistance,  gravity,  etc.,  always  have  in 
some  way  influenced  the  motion  of  the  body. 

Newton's  second  law  of  motion  says  that  a 
change  in  the  motion  of  a  body  is  proportional  to 
the  force  causing  the  change,  and  takes  place  in 
the  direction  in  which  the  force  acts.  If  several 
forces  act  on  a  body,  the  change  is  proportional  to 


140         SELF-TAUGHT  MECHANICAL  DRAWING 

the  resultant  of  the  several  forces,  and  takes  place 
in  the  direction  of  the  resultant.  This  has  been 
clearly  explained  in  the  previous  pages,  in  connec- 
tion with  the  resolution  and  composition  of  forces. 
The  most  important  point  to  note  in  regard  to 
the  second  law  of  motion  is  that  when  two  or 
more  forces  act  on  a  body  at  the  same  time,  each 
causes  a  motion  exactly  the  same  as  if  it  acted 
alone ;  each  force  produces  its  effect  independently, 
but  the  total  effect  on  the  motion  of  the  body,  of 
course,  is  a  combination  of  all  these  independent 
motions. 

Newton's  third  law  says  that  for  every  action 
there  is  an  equal  reaction.  This  means  that  if  a 
force  or  weight  presses  downward  on  a  support 
with  a  certain  pressure,  the  reaction,  or  resistance 
in  the  support,  must  equal  the  same  pressure.  If 
a  bullet  is  shot  from  a  rifle  with  a  certain  force, 
there  is  a  reaction,  or  "recoil,"  in  the  rifle,  equal 
to  the  force  required  to  give  the  velocity  to  the 
bullet.  This  law  is  very  important,  and  many 
failures  in  machine  design  have  been  due  to 
ignorance  of  the  real  meaning  of  the  law  of  action 
and  reaction. 

Newton's  third  law  may  be  illustrated  by  a  loco- 
motive drawing  a  train  of  cars.  The  driving 
wheels  give  as  much  of  a  backward  push  on  the 
rails  as  there  is  of  forward  pull  exerted  on  the 
train;  and  it  is  only  because  the  rails  are  held  in 
place  by  their  fastenings,  and  by  the  weight  rest- 
ing on  them,  that  the  locomotive  is  able  to  pull  the 
train  forward.  This  principle  of  action  and  reac- 
tion being  equal  and  opposite  is  also  an  effectual 


ELEMENTS  OF  MECHANICS  141 

bar  to  any  perpetual-motion  machine,  as  such  a 
machine  in  order  to  work  would  have  to  produce 
a  greater  action  in  one  direction  than  the  reaction 
in  the  other  direction. 

The  Pendulum. — A  body  or  weight  suspended 
from  a  fixed  point  by  a  string  or  rod,  and  free  to 
oscillate  back  and  forth  is  called  a  pendulum.  The 
center  of  oscillation  is  the  point  which,  if  all  of  the 
material  composing  the  pendulum,  including  the 
sustaining  string  or  rod,  were  concentrated  at  it 
(the  material  so  concentrated  being  considered  as 
being  suspended  by  a  line  of  no  weight)  would 
vibrate  in  the  same  time  as  the  actual  pendulum. 
The  length  of  the  pendulum  is  the  length  from  the 
point  of  suspension  to  the  center  of  oscillation. 

When  the  length  of  the  pendulum  is  unchanged, 
its  time  of  vibration  will  be  the  same,  if  its  angle 
of  vibration  does  not  exceed  three  or  four  degrees, 
and  its  time  of  vibration  will  be  but  slightly  in- 
creased for  larger  angles. 

The  time  of  vibration  of  a  pendulum  is  not 
affected  by  the  material  of  which  it  is  made, 
whether  light  or  heavy,  except  as  the  light  mate- 
rial will  offer  greater  resistance  to  the  air,  by 
presenting  a  greater  surface  in  proportion  to  its 
weight,  than  a  heavy  material. 

The  time  of  vibration  of  a  pendulum  of  a  given 
length  is  inversely  as  the  square  root  of  the  inten- 
sity of  gravity.  As  the  intensity  of  gravity  de- 
creases with  the  distance  from  the  center  of  the 
earth  it  follows  that  a  pendulum  will  vibrate  faster 
at  the  poles  or  at  sea  level  than  it  will  at  the  equa- 
tor or  at  an  elevation. 


142        SELF-TAUGHT  MECHANICAL  DRAWING 

The  time  of  vibration  of  a  pendulum  varies  di- 
rectly as  the  square  root  of  its  length.  That  is,  a 
pendulum  to  vibrate  in  one-half  or  one-third  the 
time  of  a  given  pendulum  will  need  to  be  only  one- 
quarter  or  one-ninth  of  its  length. 

Example  l.—A  pendulum  in  the  latitude  of  New 
York  will  require  to  be  39.1017  inches  long  to  beat 
seconds.  Required  the  length  of  a  pendulum  to 
make  100  beats  per  minute. 

A  pendulum  to  make  100  beats  per  minute  will 
have  to  make  its  vibrations  in  60-100  of  the  time 
of  one  which  is  making  60  beats  per  minute,  and 
its  length  will  be  equal  to  the  length  of  one  which 
beats  seconds,  multiplied  by  the  square  of  60-100, 
or: 

39.1017  X  60 2      39.1017  X  3600  , 

~W~  10,000         : 14076  mches- 

Example  #.— Required  the  time  of  vibration  of 
a  pendulum  120  inches  long.  Letting  x  repre- 
sent  the  required  time,  we  have  the  proportion 
V 120  :V 39. 1017  =  x  :  1,  or  10.954  :  6.253  =  x  :  1. 

10.954  , 

x  =  /?  oco   =  1-75  second. 
b.Zoo 

A  short  pendulum  may  be  made  to  vibrate  as 
slowly  as  desired  by  having  a  second  "bob"  placed 
above  the  point  of  suspension,  which  will  partially 
counteract  the  weight  of  the  lower  bob. 

Falling  Bodies. — A  falling  body  will  have  ac- 
quired a  velocity  at  the  end  of  the  first  second  of 
32.16  feet  per  second,  under  ordinary  conditions. 
If  the  body  is  of  such  shape  or  material  as  to  pre- 
sent a  large  surface  to  the  air  in  proportion  to  its 


ELEMENTS  OF  MECHANICS  143 

weight,  its  velocity  will,  of  course,  be  lessened,  and 
as  its  velocity  depends  upon  the  force  of  gravity, 
its  velocity  will  be  affected  somewhat  by  the  lati- 
tude of  the  place,  and  its  distance  above  sea  level. 
During  the  next  second  it  will  acquire  32.16  feet 
additional  velocity,  giving  it  a  velocity  of  64.32 
feet  at  the  end  of  the  second  second.  Each  suc- 
ceeding second  will  add  32.16  feet  to  the  velocity 
the  body  had  at  the  end  of  the  preceding  second. 

To  find  the  velocity  of  a  falling  body  at  the  end 
of  any  number  of  seconds,  therefore,  multiply  the 
number  of  seconds  during  which  the  body  has 
fallen  by  32.16.  This  rule,  expressed  as  a  formula, 
would  be : 

v  =  32.16  X  * 

in  which  v  =  velocity  in  feet  per  second,  t  =  time 
in  seconds. 

The  acceleration  due  to  gravity,  32.16  feet,  is 
often,  in  formulas,  designated  by  the  letter  g.  As 
an  example,  find  the  velocity  of  a  falling  body  at 
the  end  of  the  twelfth  second: 

v  =  32. 16X12  =  385. 92  feet. 

As  the  body  falling  starts  from  a  state  of  rest, 
its  average  velocity  will  be  one-half  of  its  final  ve- 
locity ;  the  distance  through  which  it  falls  equals 
the  average  velocity  multiplied  by  the  number  of 
seconds  during  which  it  has  been  falling.  This 
rule,  expressed  as  a  formula,  is : 

A-f  X*  .^ 

in  which  h  =  distance  or  height  through  which 


144        SELF-TAUGHT  MECHANICAL  DRAWING 

body  falls,  and  v  and  t  have  the  significance  given 
above.  But  v  =  36.16  X  t;  if  this  value  of  v  is  in- 
serted in  the  formula  just  given,  we  have: 


This  last  formula,  expressed  in  words,  gives  us 
the  rule  that  the  distance  through  which  a  body 
falls  in  a  given  time  equals  the  square  of  the  num- 
ber of  seconds  during  which  the  body  has  fallen, 
multiplied  by  16.08. 

How  long  a  distance  will  a  body  fall  in  10  sec- 
onds? Inserting  t  =  10  in  the  formula,  we  have: 

h  =  16.08  f  =  16.08  X  102  =  16.08  X  100  =  1608 
feet. 

The  time,  in  seconds,  required  for  a  body  to  fall 
a  given  distance  equals  the  square  root  of  the 
distance,  expressed  in  feet,  divided  by  4.01.  Ex- 
pressed as  a  formula,  this  rule  would  be  : 

t  =  V^ 
4.01' 

As  an  example,  assume  that  a  stone  falls  through 
a  distance  of  3600  feet.  How  long  time  is  required 
for  this? 

Inserting  h  =  3600  in  the  formula,  we  have  : 

V3600         60 
t  =      .          =          =  15  seconds,  very  nearly. 


The  velocity  of  a  falling  body  after  it  has  fallen 
through  a  given  distance  equals  the  square  root  of 
the  distance  through  which  it  has  fallen  multi- 
plied by  8.02. 


ELEMENTS  OF  MECHANICS  145 

This  rule,  expressed  as  a  formula,  is: 


What  is  the  velocity  of  a  falling  body  after  it 
has  fallen  through  a  distance  of  3600  feet? 
Inserting  h  =  3600  in  the  formula,  we  have: 

v  =  8.02  X  V3600  =  8.02  X  60  =  481.2  feet. 

The  height  from  which  a  body  must  fall  to  acquire 
a  given  velocity  equals  the  square  of  the  velocity 
divided  by  64.32.  As  a  formula,  this  rule  is: 


~  64.32 

From  what  height  must  a  body  fall  to  acquire  a 
velocity  of  500  feet  per  second?  Inserting  v  =  500 
in  the  formula  given,  we  have: 

500  2       500  X  500 
=  64.32  =       .64.32 

If  a  body  is  thrown  upward  with  a  given  ve- 
locity, its  velocity  will  diminish  during  each  second 
at  the  same  rate  as  it  increases  when  the  body 
falls.  A  body  thrown  up  into  the  air  in  a  vertical 
direction  will  return  to  the  ground  with  exactly 
the  same  velocity  as  that  with  which  it  was  thrown 
into  the  air.  At  any  point,  the  velocity  on  the  up- 
ward journey  will  be  equal  to  the  velocity  on  the 
downward  journey,  except  that  the  direction  is 
reversed. 

The  acceleration  of  a  falling  body,  32.16  feet  per 
second,  is  the  value  at  the  latitude  of  New  York, 
at  sea  level. 

The  force  required  to  give  to  a  falling  body  its 


146        SELF-TAUGHT  MECHANICAL  DRAWING 

acceleration  of  32.16  feet  per  second  is  the  weight 
of  the  body  itself.  The  force  required  to  give  any 
acceleration  to  a  body,  then,  is  to  the  weight  of  the 
body  as  that  acceleration  is  to  the  acceleration 
produced  by  gravity.  Therefore,  to  find  the  force 
required  to  produce  a  given  rate  of  acceleration  to 
a  body,  divide  the  weight  of  the  body  by  32.16, 
and  multiply  the  quotient  by  the  required  rate  of 
acceleration. 

Example. — A  body  weighing  125  pounds  is  to  be 
lifted  with  an  acceleration  of  10  feet  per  second. 
Required  the  strain  on  the  sustaining  rope. 

125 

00  .,„  X  10  =  38.8,  the  tension  necessary  to  produce 
oZ.lb 

the  acceleration. 

To  this  must  be  added  the  pull  necessary  to  lift 
the  weight  without  acceleration,  or  the  weight  of 
the  body  itself.  Thus  38.8  +  125  =  163.8  is  the  re- 
quired tension  on  the  rope. 

The  rate  of  acceleration  which  a  continuously 
acting  force  will  produce  is  equal  to  the  force 
divided  by  the  weight  of  the  body,,  multiplied  by 
32.16. 

Energy  and  Work. — The  unit  of  work,  the  stand- 
ard by  which  work  is  measured,  is  the  foot-pound, 
or  the  amount  of  work  done  in  lifting  a  weight  or 
overcoming  a  resistance  of  one  pound  through  one 
foot  of  space. 

"Energy  is  the  product  of  a  force  factor  and  a 
space  factor.  Energy  per  unit  of  time,  or  rate  of 
doing  work,  is  the  product  of  a  force  factor  and  a 
velocity  factor,  since  velocity  is  space  per  unit  of 
time.  Either  factor  may  be  changed  at  the  ex- 


ELEMENTS  OF  MECHANICS  147 

pense  of  the  other;  i.e.,  velocity  may  be  changed, 
if  accompanied  by  such  a  change  of  force  that  the 
energy  per  unit  of  time  remains  constant.  Corre- 
spondingly force  may  be  changed  at  the  expense  of 
velocity,  energy  per  unit  of  time  being  constant. 
Example. — A  belt  transmits  6000  foot-pounds  per 
minute  to  a  machine.  The  belt  velocity  is  120  feet 
per  minute,  and  the  force  exerted  is  50  pounds. 
Frictional  resistance  is  neglected.  A  cutting  tool 
in  the  machine  does  useful  work ;  its  velocity  is  20 
feet  per  minute,  and  the  resistance  to  cutting  is 
300  pounds.  Then  the  energy  received  per  minute 
-  120  X  50  =  6000  foot-pounds;  and  energy  deliv- 
ered per  minute  =  20  X  300  =  6000  foot-pounds. 
The  energy  received  therefore  equals  the  energy 
delivered.  But  the  velocity  and  force  factors  are 
quite  different  in  the  two  cases."  (Prof.  A.  W. 
Smith.) 

Force  of  the  Blow  of  a  Steam  Hammer  or  Other 
Falling  Weight— The  question,  "With  what  force 
does  a  falling  hammer  strike  ?"  is  often  asked. 
This  question  can,  however,  not  be  answered 
directly.  The  energy  of  a  falling  body  cannot  be. 
expressed  in  pounds,  simply,  but  must  be  expressed 
in  foot-pounds.  The  energy  equals  the  weight  of 
the  falling  body  multiplied  by  the  distance  through 
which  it  falls,  or,  expressed  as  a  formula: 

E  =  WXh, 

in  which  E  =  energy  in  foot-pounds, 

W  =  weight  of  falling  body  in  pounds, 
h  =  height  from  which  body  falls  in  feet. 

The  energy  can  also  be  found  by  dividing  the 


148         SELF-TAUGHT  MECHANICAL  DRAWING 

weight  of  the  falling  body  by  64.32  and  then  mul- 
tiplying the  quotient  by  the  square  of  the  velocity 
at  the  end  of  the  distance  through  which  it  falls. 
This  rule,  expressed  as  a  formula,  is: 


in  which  E  and  W  denote  the  same  quantities  as 
before,  and  v  =  the  velocity  of  the  body  at  the  end 
of  its  fall. 

Both  of  these  formulas  give,  of  course,  the  same 
results.  That  the  second  method  gives  the  same 
result  as  multiplying  the  weight  by  the  height 
through  which  it  falls,  is  evident  from  the  fact, 
stated  under  the  head  of  "Falling  Bodies/'  that  the 
square  of.  the  velocity  of  a  falling  body,  divided 
by  64.32,  gives  the  height  through  which  it  has 
fallen. 

This  second  method  allows  of  determining  the 
energy  of  any  weight  or  force  moving  at  a  given 
velocity,  whether  its  velocity  has  been  acquired  by 
falling,  or  is  due  to  other  causes. 

Now  assume  that  we  wish  to  find  the  force  of 
the  blow  of  a  300-pound  drop  hammer,  falling  2 
feet  before  striking  the  forging,  and  compressing 
it  2  inches. 

The  energy  of  the  falling  hammer  when  reach- 
ing the  forging  is: 

E  =  W  X  h  =  300  X  2  =  600  foot-pounds. 

This  energy  is  used  during  the  act  of  compress- 
ing the  forging  2  inches  or  0.166  of  a  foot.  Con- 
sequently, the  average  force  of  the  hammer  with 


ELEMENTS  OF  MECHANICS  149 

which  it  compresses  the  forging  is  600  -*-  0.166  + 
the  weight  of  the  hammer,  or 


Average  force  of  blow  =  A  «„,*  +  300  = 

U.lbb 

3600  +  300  =  3900  pounds. 
The  general  formula  for  the  force  of  a  blow  is: 


in  which  F  =  average  force  of  blow  in  pounds, 
W  =  weight  of  hammer  in  pounds, 
h  =  height  of  drop  of  hammer  in  feet, 
d  =  penetration  of  blow  in  feet. 

A  horse-power,  in  mechanics,  is  the  power  ex- 
erted, or  work  done,  in  lifting  a  weight  of  33,000 
pounds  one  foot  per  minute,  or  550  pounds  one  foot 
per  second.  The  power  exerted  by  a  piston  driven 
by  steam  or  other  medium  during  one  stroke,  in 
foot-pounds,  is  equal  to  the  area  of  the  piston, 
multiplied  by  the  pressure  per  square  inch,  multi- 
plied by  the  stroke  in  feet,  the  product  of  the  area 
by  the  pressure  giving  the  force,  and  the  stroke 
giving  the  distance  through  which  the  force  is 
exerted.  In  the  case  of  steam  engines,  where  the 
steam  is  cut  off  at  one-quarter,  one-  third  or  one- 
half  of  the  stroke,  the  piston  being  driven  the  rest 
of  the  way  by  the  expansion  of  the  steam,  the 
average  pressure  for  the  entire  stroke,  the  ''mean 
effective  pressure"  (M.E.  P.),  as  it  is  called,  is  the 
basis  of  calculations.  As  each  revolution  of  the 
engine  equals  two  strokes  of  the  piston,  the  number 
of  foot-pounds  per  minute  an  engine  is  developing 
will  be  the  product  of  the  area  of  the  piston  in 


150         SELF-TAUGHT  MECHANICAL  DRAWING 

square  inches,  multiplied  by  the  mean  effective 
pressure,  multiplied  by  the  stroke  in  feet,  multi- 
plied by  the  number  of  revolutions   per  minute 
times  2.    This  product,  divided  by  33,000,  gives 
the  indicated  horse-power  (I.H.P.)  of  the  engine; 
this  name  being  derived  from  the  fact  that  the 
mean  effective  pressure  is  determined  by  the  use 
of  the  steam  engine  indicator.    Therefore: 
r  TT  D     Area  XM.  E. P.  X  stroke  X  rev,  permin.  X2 
LH'P'  =  33,000 

This  formula  may  be  transposed  in  various  ways 
to  give  other  information.  For  instance,  if  the 
piston  area  for  a  given  horse-power  is  desired, 
then 

Area  LH.P.  X  33,000 

M.E.P.  X  stroke  X  rev.  per  min.  X  2. 

If  the  volume  of  the  cylinder  is  desired,  then 

A  LH.P.  X  33000 

Area  X  stroke  =  , ,  ^  n  — — rr^r 

M.E.P.  X  rev.  permin.  X  2. 

If  the  pressure  to  produce  a  given  horse-power 
is  desired,  then 

MEp   = LH.P.  X  33000 

Area  X  stroke  X  rev.  per  min.  X  2. 

The  mean  effective  pressure  in  the  cylinder  of 
the  engine  is,  of  course,  considerably  less  than  the 
boiler  pressure  as  shown  by  the  steam  gauge.  The 
indicated  horse-power  of  an  engine  does  not  take 
into  account  the  losses  caused  by  the  friction  of 
the  working  parts.  The  power  which  the  engine 
actually  delivers  as  shown  by  a  brake  dynamo- 
meter or  other  contrivance  at  the  flywheel  is  called 
the  brake  horse-power. 


CHAPTER  IX 

FIRST  PRINCIPLES  OF  STRENGTH  OF  MATERIALS 

Factor  of  Safety. — It  is  obvious  that  it  would  be 
unsafe  in  designing  a  piece  of  construction  work 
to  allow  a  strain  of  anywhere  near  the  breaking 
limit  of  the  material  it  is  to  be  made  from.  It  is, 
therefore,  customary  in  making  any  calculations 
for  the  size  of  the  parts  to  use  what  is  called  a 
factor  of  safety,  by  making  the  part  from  three  or 
four  to  ten  or  even  more  times  the  strength  neces- 
sary to  just  resist  breaking  with  a  steady  load. 
The  factor  of  safety  used  will  depend  upon  several 
considerations.  It  will  depend,  first,  upon  the  na- 
ture of  the  material  used.  A  wrought  or  drawn 
metal,  for  instance,  will  be  likely  to  be  more  uni- 
form in  its  nature  than  a  cast  metal  which  may 
contain  air  holes,  or  which  may  be  more  or  less 
spongy,  or  which  may  be  under  unequal  strains  in 
cooling.  The  matter  of  strains  in  a  casting  due  to 
unequal  cooling  is  to  a  considerable  extent  a  mat- 
ter of  proper  or  improper  design ;  still  it  is  not 
possible  to  entirely  avoid  them. 

Again  the  factor  of  safety  to  be  used  will  depend 
upon  the  nature  of  the  work  which  will  be  re- 
quired of  the  part.  If  the  part  has  to  simply  sus- 
tain a  steady  load  it  will  not  need  to  be  as  strong 
as  though  the  load  was  applied  and  reversed,  or 

151 


152         SELF-TAUGHT  MECHANICAL  DRAWING 

even  as  strong  as  though  the  load  was  applied  and 
released.  To  illustrate,  it  is  a  familiar  fact  that  a 
piece  of  wire  which  may  be  bent  a  given  amount 
without  apparent  injury,  may  be  broken  by  repeat- 
edly bending  it  back  and  forth  the  same  amount 
at  one  point.  And,  similarly,  in  machine  parts, 
rupture  may  be  caused  not  only  by  a  steady  load 
which  exceeds  the  carrying  strength,  but  by  re- 
peated applications  of  stresses  none  of  which  are 
equal  to  the  carrying  strength.  Rupture  may  also 
be  caused  by  a  succession  of  shocks  or  impacts, 
none  of  which  alone  would  be  sufficient  to  cause  it. 
Iron  axles,  the  piston  rods  of  steam  hammers  and 
other  pieces  of  metal  subjected  to  repeated  shocks, 
invariably  break  after  a  certain  length  of  service. 

The  factor  of  safety  used  will  therefore  vary 
widely  with  the  nature  of  the  work  required  of  the 
part.  For  a  steady  or  "dead"  load,  Prof.  A.  W. 
Smith  says:  "In  exceptional  cases  where  the 
stresses  permit  of  accurate  calculation,  and  the 
material  is  of  proven  high  grade  and  positively 
known  strength,  the  factor  of  safety  has  been 
given  as  low  a  value  as  1J  •  but  values  of  2  and  3  are 
ordinarily  used  for  iron  or  steel  free  from  welds ; 
while  4  to  5  are  as  small  as  should  be  used  for  cast 
iron  on  account  of  the  uncertainty  of  its  composi- 
tion, the  danger  of  sponginess  of  structure,  and 
indeterminate  shrinkage  stresses."  Others  would 
make  3  the  lowest  factor  of  safety  that  should  be 
used  for  wrought  iron  and  steel. 

Where  the  load  is  variable,  but  well  within  the 
elastic  limit  of  the  material,  that  is  where  the  load 
is  not  so  great  but  so  that  the  part  will  immedi- 


STRENGTH  OF  MATERIALS  153 

ately  resume  its  original  shape  when  the  load  is 
removed,  a  factor  of  safety  of  5  or  6  might  be 
used.  The  part  will  need  to  be  made  stronger  if 
the  load  or  force  acts  first  in  one  direction  and 
then  in  the  opposite  direction,  that  is,  if  it  acts 
back  and  forth,  than  it  will  need  to  be  if  the  same 
force  is  simply  applied  and  then  released.  Where 
the  part  is  subjected  to  shock,  the  factor  of  safety 
is  generally  made  not  less  than  10.  A  factor  of 
safety  as  high  as  40  has  been  used  for  shafts  in 
mill-work  which  transmit  very  variable  powers. 

In  cases  where  the  forces  are  of  such  a  nature 
that  they  cannot  be  determined,  then  Prof.  Smith 
says:  "Appeal  must  be  made  to  the  precedent  of 
successful  practice,  or  to  the  judgment  of  some  ex- 
perienced man  until  one's  own  judgment  becomes 
trustworthy  by  experience.  *  *  *  In  proportioning 
machine  parts,  the  designer  must  always  be  sure 
that  the  stress  which  is  the  basis  of  calculation 
or  the  estimate,  is  the  maximum  possible  stress; 
otherwise  the  part  will  be  incorrectly  propor- 
tioned." And  he  cites  the  case  of  a  pulley  where 
if  the  arms  were  to  be  designed  only  to  resist  the 
belt  tension  they  would  be  absurdly  small,  because 
the  stresses  resulting  from  the  shrinkage  of  the 
casting  in  cooling  are  often  far  greater  than  those 
due  to  the  belt  pull. 

In  many  cases  the  practical  question  of  feasi- 
bility of  casting  will  determine  the  thickness  of 
parts,  independent  of  the  question  of  strength. 
For  instance,  on  small  brass  work,  such  as  plumb- 
ers' supply,  and  small  valve  work,  a  thickness  of 
about  3-32  of  an  inch  is  as  little  as  can  be  relied 


154         SELF-TAUGHT  MECHANICAL  DRAWING 

on  to  make  a  good  casting  on  cored  out  work ;  or 
in  the  case  of  partitions  in  such  work  where  the 
metal  has  to  flow  in  between  cores,  a  thickness  of 
about  J  of  an  inch  is  as  small  as  should  be  used; 
yet  such  thicknesses  may  be  much  greater  than 
are  required  to  give  the  necessary  strength.  On 
larger  cast  iron  work,  the  thickness  to  be  allowed 
to  insure  a  good  casting  will,  of  course,  depend 
upon  the  size  of  the  piece.  The  judgment  of  the 
pattern-maker  or  foundry-man  will  naturally  de- 
termine the  thickness  in  such  cases. 

Shape  of  Machine  Parts. — While  the  size  of  ma- 
chine parts  will  vary  greatly  with  the  nature  of  the 
work  required  of  them,  their  shape  will  depend 
very  much  on  the  manner  or  direction  in  which 
the  load  or  strain  is  brought  to  bear  upon  them. 
If  the  part  is  subjected  to  simple  tension,  that  is, 
merely  resists  a  force  tending  to  pull  it  apart,  then 
the  shape  of  the  member  which  serves  this  purpose 
is  not  very  material,  though  a  round  rod,  being  most 
compact  and  cheapest,  is  best.  Almost  any  shape 
will  answer,  however,  though  it  is  well  to  avoid 
using  thin  and  broad  parts,  as  a  strain,  though  not 
greater  than  that  which  the  part  as  a  whole  might 
bear  safely,  might  be  brought  upon  one  edge,  pro- 
ducing^a  tearing  effect  beyond  the  safe  limit.  For 
resisting  simple  tension  the  part  should  be  made  of 
uniform  size  its  entire  length,  of  a  size  to  be  deter- 
mined by  the  tensile  strength  of  the  material  and 
the  factor  of  safety  used. 

If  the  part  is  to  resist  compression,  then  when 
the  proportion  of  its  length  to  its  diameter  or 
thickness  is  such  that  it  will  "  buckle' '  or  bend, 


STRENGTH  OF  MATERIALS  155 

instead  of  crushing,  that  is  when  its  length  ex- 
ceeds five  or  six  times  its  diameter,  it  becomes 
desirable  to  use  a  hollow  or  cross-ribbed  form  of 
construction,  so  as  to  get  the  metal  as  far  from  the 
axis  of  the  piece  as  possible.  The  hollow  cylind- 
rical form,  by  getting  all  of  the  metal  equally  dis- 
tant from  the  axis  is,  of  course,  most  effective, 
but  considerations  of  appearance  may  make  a  hol- 
low square  form  more  desirable,  while  considera- 
tions of  cost  may  make  a  cross-ribbed  form  to  be 
preferred,  as  such  a  form  can  be  cast  without  the 
use  of  cores.  In  cases  where  a  wrought  metal  must 
be  used  a  solid  form  is  often  the  only  practicable 
one.  When  it  becomes  important  to  keep  the 
weight  down  to  the  lowest  point,  it  is  common  to 
have  the  piece  slightly  enlarged  in  the  middle 
of  its  length,  as  in  the  case  of  connecting  rods  of 
steam  engines.  In  the  case  of  steam  engine  con- 
necting-rods, the  tendency  to  buckle  is  least  side- 
ways, as  the  cross-head  and  crank-pins  tend  to 
hold  it  in  line  this  way,  while  the  rotary  motion  of 
the  crank-pin  tends  to  produce  buckling  the  other 
way.  Connecting  rods  are  therefore  frequently 
made  somewhat  flat,  of  a  breadth  about  twice 
their  thickness. 

When  a  piece  is  designed  to  resist  bending,  it 
becomes  desirable  to  get  a  good  depth  of  material 
in  the  direction  in  which  the  force  is  applied,  as 
the  capacity  of  a  piece  to  resist  bending  increases 
as  the  square  of  its  thickness  or  depth  in  the  di- 
rection of  the  force,  but  only  directly  as  its  breadth 
or  width,  so  that  to  increase  the  thickness  of  a 
piece  two  or  three  times  in  the  direction  of  the 


156        SELF-TAUGHT  MECHANICAL  DRAWING 

force  would  increase  its  capacity  to  resist  bending 
four  or  nine  times;  while  to  increase  its  breadth 
two  or  three  times  would  only  increase  its  strength 
two  or  three  times.  The  proportion  of  depth  to 
breadth  which  can  be  used  will,  of  course,  depend 
upon  the  length  of  the  piece,  as  if  the  piece  is  long 
and  its  depth  is  made  large  in  proportion  to  its 
thickness  the  tendency  will  be  for  the  piece  to 
buckle,  or  yield  sideways.  To  resist  this  tendency 
it  is  customary  to  put  ribs  on  the  edges  of  such  a 


FIG.  130.  FIG.  131. 

FIGS.  130  and  131.— Beam  Cross-sections  of 
Different  Types. 

piece,  giving  it  the  form  shown  in  Fig.  130.  The 
hollow  box-form  shown  in  Fig.  131  is,  of  course, 
equally  effective  to  resist  combined  bending  and 
buckling  stresses,  and  in  some  cases  may  be  pref- 
erable as  a  matter  of  appearance  on  account  of 
the  impression  of  solidity  which  it  gives. 

A  projecting  beam,  like  that  shown  in  Fig.  132, 
designed  to  resist  a  force  or  sustain  a  load  at 
its  end,  would  need  to  have  its  lower  edge  made 
of  the  form  of  a  parabola,  if  made  of  uniform 
thickness.  If  the  edges  were  ribbed  to  prevent 
buckling,  then  material  might  be  taken  out  of 
the  middle  portion,  as  shown  in  Fig.  133,  without 
weakening  it. 


STRENGTH  OF  MATERIALS 


157 


Strength  of  Materials  as  Given  by  Kirkaldy's 
Tests. — A  very  large  number  of  tests  of  cast  iron 
made  by  Kirkaldy  gave  results  as  follows :  Tensile 
strength  per  square  inch,  necessary  to  just  tear 
asunder,  from  about  10,000  or  12,000  pounds  to 
about  28,000  or  32,000  pounds,  or  an  average 
strength  of  about  20,000  pounds.  Tests  on  the 
ability  of  cast  iron  to  resist  crushing  gave  results 
vary  ing  from  about  50, 000  to  about  150, 000  pounds, 


FIG.  132.  —  Cantilever  of 
Uniform  Strength,  when 
Loaded  at  End. 


FIG.  133.  — Common  Design 
of  Cantilever  of  Uniform 
Strength. 


or  an  average  strength  of  about  100,000  pounds 
per  square  inch.  These  tests  indicate  that  cast 
iron  has  about  five  times  the  capacity  to  resist 
crushing  that  it  has  to  resist  tension.  They  also 
indicate  that  cast  iron  is  a  somewhat  uncertain 
material. 

Tests  of  wrought  iron  indicated  a  tensile  strength 
of  between  40,000  and  50,000  pounds  per  square 
inch,  the  elastic  limit  being  reached  at  about  one- 
half  the  tensile  strength.  Tests  on  steel  castings 
gave  results  for  tensile  strength  ranging  from 
55,000  to  about  64,000  pounds  per  square  inch, 


158         SELF-TAUGHT  MECHANICAL  DRAWING 

the  elastic  limit  being  reached  at  about  30,000 
pounds. 

Tests  of  wire  gave  results  as  follows :  Brass, 
from  81,000  to  98,000  pounds  per  square  inch  of 
area.  Iron,  from  59,000  to  97,000  pounds.  Steel, 
from  103,000  to  318,000  pounds. 

The  tensile  strength  of  regular  machine  steel 
(low  carbon  steel)  is  generally  given  at  about 
60,000  pounds  per  square  inch. 

Size  of  Parts  to  Resist  Stresses. — To  resist  ten- 
sion it  is,  of  course,  only  necessary  to  have  the 
piece  of  such  a  size  that  each  square  inch  shall  not 
have  a  stress  greater  than  the  average  strength 
of  the  material  (as  20,000  pounds  for  cast  iron) 
divided  by  whatever  factor  of  safety  may  be 
selected. 

To  Resist  Crushing.— Prof.  Hodgkinson's  rule 
for  the  strength  of  hollow  cast  iron  pillars  is  as 
follows :  To  ascertain  the  crushing  weight  in  tons 
multiply  the  outside  diameter  by  3.55;  from  this 
subtract  the  product  of  the  inside  diameter  multi- 
plied by  3.55,  and  divide  by  the  length  multiplied 
by  1.7.  Multiply  this  quotient  by  "46.65.  Ex- 
pressed as  a  formula  this  rule  would  be: 

(D  X  3.55)  -  (d  X  3. 55) 
L  X  1.7 


Sc  =  46.65  X 


in  which 
Sc  =  ultimate  compressive  (crushing)   strength 

of  hollow  column,  in  tons, 
D  =  outside  diameter  in  inches, 
d  =  inside  diameter  in  inches, 
L  =  length  of  column  in  feet. 


STRENGTH  OF  MATERIALS 


159 


Any  desired  factor  of  safety  may  be  introduced 
in  the  above  formula  by  dividing  the  factor  46.65 
by  the  factor  of  safety.  In  this  case  the  formula 
would  be: 

Q        46J>5_Xl CD ._X_SL55)  -  (d  X  3.55)] 
F  X  L  X  1.7 

in  which 

S«  =  safe  compressive  strength  in  tons, 

F  =  factor  of  safety,  and 

D,  d  and  L  have  the  same  meaning  as  above. 

This  rule  and  formula  assumes  that  the  ends  of 
the  column  are  perfectly  flat  and  square,  and  that 
the  load  bears  evenly  on  the  whole  surface. 

If  the  ends  are  rounded,  the  column  yields  at 
about  one-half  the -stress  of  one  with  fixed  square 
ends. 

To  Resist  Bending. — In  the  following  commonly 
given  rules  for  the  strength  of  beams  or  bars  to 


— -i 


K-H 


I 


FIG.  134.— Rectangular  Cantilever. 


resist  breaking  by  transverse  stresses,  the  tensile 
strength  of  cast  iron  is  assumed  at  20,000  pounds 
per  square  inch.  Divide  20,000  in  the  formulas 


> 


160        SELF-TAUGHT  MECHANICAL  DRAWING 

by  the  desired  factor  of  safety.  The  breadth  and 
depth  of  rectangular  bars,  the  diameter,  if  the  bar 
is  round,  and  the  length,  are  all  in  inches. 

For  rectangular  bars  fixed  at  one  end  with  the 
force  applied  at  the  other,  Fig.  134,  the  breaking 
load  equals 

bX  d2X  20,000 
I 


: 


For  round  bars  under  the  same  conditions,  Fig. 
135,  the  breaking  load  equals 

JL    /  0.59  X  d*X  20,000 
6  I 

If  the  rectangular  bar  is  hollow,  as  shown  in 


FIG.  135. — Circular  Section  Cantilever. 


Fig.  131,  subtract  the  internal  b  X  d2  from  the 
external  b  X  d2. 

If  the  round  bar  is  hollow  subtract  the  internal 
d 3  from  the  external  d 3. 

The  case  of  a  bar  of  the  I-section  shown  in  Fig. 
130  is  similar  to  that  of  the  hollow  rectangular  bar 
of  Fig.  131,  the  depressions  in  its  sides  correspond- 
ing to  the  hollow  part  of  Fig.  131,  the  sum  of  their 


STRENGTH  OF  MATERIALS  161 

depths  corresponding  with  the  internal  width  b  of 
the  hollow  rectangular  bar. 

If  a  beam  is  fixed  at  one  end  and  the  load  is 
evenly  distributed  throughout  its  entire  length, 
it  will  bear  double  the  weight  it  will  if  the  load 
is  supported  at  the  outer  end. 

If  the  beam  is  supported  at  the  ends  and  loaded 
in  the  middle  it  will  bear  four  times  the  weight  of 
the  beam  of  Fig.  134,  or,  if  the  load  is  evenly  dis- 
tributed throughout  the  length  of  the  beam,  eight 
times. 

If  the  beam,  instead  of  being  simply  supported 
at- the  ends,  has  the  ends  fixed  and  is  loaded  at  the 
center,  its  ability  to  resist  breaking  will  be  doubled 
as  compared  with  that  when  loaded  at  the  center 
and  with  the  ends  only  supported. 

Regarding  the  safe  load  that  beams  or  bars  of 
different  material  may  bear  Griffin  says  that  "with 
but  a  general  knowledge  of  the  elastic  limit,  ordi- 
nary steel  is  good  for  from  between  12,000  to  15,000 
pounds  per  square  inch  non-reversing  stress,  and 
from  8000  to  10,000  pounds  reversing  stress.  Cast 
iron  is  such  an  uncertain  metal,  on  account  of  its 
variable  structure,  that  stresses  are  always  kept 
low,  say  from  3000  to  4000  for  non-reversing  stress, 
and  1500  to  2500  for  reversing  stress. " 

Again,  though  the  tests  of  wrought  iron  show  it 
to  have  a  much  higher  tensile  strength  than  cast 
iron,  Nystrom,  in  formulas  for  lateral  strength, 
gives  wrought  iron  but  little  more  than  three-quar- 
ters the  value  of  cast  iron,  probably  because  it 
bends  so  readily. 


162 


SELF-TAUGHT  MECHANICAL  DRAWING 


A  table  is  appended  giving  the  average  breaking 
strength,  in  pounds  per  square  inch,  of  some  com- 
monly used  materials  in  engineering  practice. 


Tension. 

Compression. 

Aluminum    

15,000 

12,000 

24,000 

30,000 

Copper  cast 

24  000 

40,000 

Iron   cast 

15,000 

80,000 

Iron   wrought       .... 

48,000 

46,000 

Steel  castings  

70,000 

70,000 

Structural  steel     .... 

60,000 

60,000 

Stresses  in  Castings. — Reference  has  been  pre- 
viously made  to  stresses  in  castings,  due  to  shrink- 
age in  cooling.  If  all  parts  of  a  casting  could  be 
made  to  cool  equally  fast  there  would  not  be  much 
trouble  in  this  respect,  but  as  different  parts  of  a 
casting  vary  in  thickness,  the  time  they  require 
to  cool  will  vary,  and  the  thick  parts  remaining 
fluid  the  longest,  will,  on  cooling,  cause  a  strain  on 
the  already  cool  thin  parts.  In  the  case  of  a  pulley, 
where  the  rim  and  arms  are  much  lighter  than  the 
hub,  the  hub  on  cooling  will  tend  to  draw  the  arms 
to  itself  and  away  from  the  rim,  and  if  the  differ- 
ence in  thickness  is  great,  they  may  be  even  found 
to  be  pulled  away  so  as  to  show  a  crack  where  they 
join  the  rim.  The"remedy  in  such  a  case  would,  of 
course,  be  first,  to  take  out  as  much  of  the  metal 
from  the  center  of  the  hub  as  possible  by  means 
of  a  core,  and  second,  to  keep  the  outside  of  the 
hub  as  small  as  would  be  consistent  with  strength, 
getting  necessary  thickness  for  set  screws  by  hav- 
ing a  raised  place  or  boss  at  that  point. 


STRENGTH  OF  MATERIALS  163 

As  these  strains  are  primarily  due  to  unequal 
cooling,  it  is  evident  that  in  order  to  reduce  them 
to  the  lowest  point  the  first  thing  to  do  is  to  make 
the  different  parts  of  the  casting  of  as  nearly  uni- 
form thickness  as  possible.  Where  different  parts 
of  the  casting  vary  in  thickness,  the  change  from 
one  thickness  to  the  other  should  be  made  as  grad- 
ual as  possible.  Sharp  internal  corners  should  also 
be  avoided,  as  such  places  are  very  liable  to  be 
spongy;  the  sand  from  the  sharp  corner  in  the 
mould  is  also  very  liable  to  wash  away  when  the 
metal  is  poured  in,  and  lodge  in  some  other  place, 
causing  a  defective  casting.  A  good  ' ' fillet, "  as  an 
internal  round  corner  is  called,  which  the  pattern- 
maker may  put  into  the  pattern  with  wax,  putty  or 
leather,  will  not  be  very  expensive,  and  will  save 
much  trouble  in  the  casting. 

Besides  possessing  a  knowledge  of  factors  of 
safety,  proportioning  parts  to  resist  various 
stresses  and  the  like,  a  general  knowledge  of  the 
principles  of  foundry  and  machine  shop  practice  is 
essential  to  properly  design  machine  work.  If  one 
does  not  understand  foundry  work,  he  will  be  con- 
stantly designing  castings  which  it  will  be  im- 
practicable to  mould ;  if  not  actually  impossible  of 
moulding,  they  will  be  needlessly  expensive.  And 
in  like  manner,  unless  he  understands  the  general 
principles  of  machine  shop  practice,  his  work  will 
be  giving  trouble  at  that  end  of  the  line. 


CHAPTER  X 

CAMS 

General  Principles. — In  designing  machinery  it 
is  frequently  desirable  to  give  to  some  part  of  the 
mechanism  an  irregular  motion.  This  is  often 
done  by  the  use  of  cams,  which  are  made  of  such 
form  that  when  they  receive  motion,  either  rotary 
or  reciprocating,  they  impart  to  a  follower  the 
desired  irregular  motion. 

The  follower  is  sometimes  flat,  and  sometimes 
round.  When  the  follower  is  round  it  is  usually 
made  in  the  form  of  a  wheel  or  roller,  so  as  to  les- 
sen the  wear  and  the  friction.  The  follower  may 
work  upon  the  edge  of  the  cam,  or  if  round,  it 
may  work  in  a  groove  formed  either  on  the  face 
or  on  the  side  of  the  cam. 

The  working  surfaces  of  cams  with  round  fol- 
lowers are  laid  out  from  a  pitch  line,  so  called, 
which  passes  through  the  center  of  the  follower. 
The  shape  of  this  pitch  line  determines  the  work 
which  the  cam  will  do.  The  working  surface  of 
the  cam  is  at  a  distance  from  the  follower  equal  to 
one-half  the  diameter  of  the  follower.  This  prin- 
ciple of  a  pitch  line  holds  good  whether  the  cam 
works  only  upon  its  edge  like  the  one  shown  in 
Fig.  139,  or  whether  it  has  an  outer  portion  to 
insure  the  positive  return  of  the  follower.  This 

164 


CAMS 


165 


outer  portion  is  frequently  made  in  the  form  of  a 
rim  of  uniform  thickness  around  the  groove. 

Design  a  Cam  Having  a  Straight  Follower  Which 
Moves  Toward  or  From  the  Axis  of  the  Cam,  as 
Shown  in  Fig.  136. — Let  it  be  required  that  the 
follower  shall  advance  at  a  uniform  rate  from  a  to 


FIG.  136.— Cam  with  Straight  Follower  having  Uniform 
Motion. 


6  as  the  cam  makes  a  half  revolution,  this  advance 
being  preceded  and  followed  by  a  period  of  rest  of 
a  twelfth  of  a  revolution  of  the  cam. 

Divide  that  half  of  the  cam  during  the  revolu- 
tion of  which  the  follower  is  to  be  raised  from  a  to 
&,  in  this  case  the  half  at  the  right  of  the  vertical 
center  line,  into  a  number  of  equal  angles,  and 


166         SELF-TAUGHT  MECHANICAL  DRAWING 

divide  the  distance  from  a  to  b  into  the  same  num- 
ber of  equal  spaces.  Mark  off  the  points  so  ob- 
tained onto  the  successive  radial  lines  as  indicated 
by  the  dotted  lines,  and  at  the  points  where  these 
dotted  lines  intersect  the  radial  lines  draw  lines  at 
right  angles  to  the  radial  lines  to  represent  the 
position  of  the  follower  when  these  radial  lines 
become  vertical  as  the  cam  revolves. 

A  period  of  rest  in  a  cam  is  represented  by  a  cir- 
cular portion,  having  the  axis  of  the  cam  as  its 
center.  In  order,  therefore,  to  obtain  the  required 
periods  of  rest,  the  distances  of  a  and  b  from  the 
center  are  marked  off  upon  the  radial  lines  c  and 
d,  these  lines  being  made  a  twelfth  of  a  revolution 
from  the  vertical  center  line,  and  lines  represent- 
ing the  follower  are  drawn  at  these  points  as  be- 
fore. To  get  the  return  of  the  follower  the  space 
from  c  to  d  is  divided  into  a  number  of  equal 
angles,  and  the  distance  from  e  to /is  divided  off 
to  represent  the  desired  rate  of  return  of  the 
follower.  In  this  case  the  rate  of  return  is  made 
uniform,  so  the  distance  ef  is  spaced  off  equally. 
The  distance  of  these  points  from  the  axis  is  marked 
off  upon  the  radial  lines  between  c  and  d,  and  lines 
representing  the  follower  are  drawn. 

A  curved  line,  which  may  be  made  with  the 
aid  of  the  irregular  curves,  which  is  tangent  to  all 
of  the  lines  representing  the  follower,  gives  the 
shape  of  the  cam. 

Fig.  137  shows  a  cam  having  the  conditions  as  to 
the  rise,  rest  and  return  of  the  follower  the  same 
as  the  one  shown  in  Fig.  136,  the  follower,  how- 
ever, being  pivoted  at  one  end. 


CAMS 


167 


Draw  the  arc  ab  representing  the  path  of  a  point 
in  the  follower  at  the  vertical  center  line,  and 
divide  that  part  of  the  arc  through  which  the  fol- 
lower rises  into  the  same  number  of  equal  spaces 
as  the  half  circle  at  the  right  of  the  vertical  cen- 
ter line  is  divided  into  angles.  Through  these 


FlG.  137.— Cam  with  Pivoted  Follower. 


points  draw  lines,  as  shown,  representing  consecu- 
tive positions  of  the  working  face  of  the  follower. 
The  various  distances  of  the  follower  from  the  axis 
of  the  cam  are  now  marked  off  upon  the  corre- 
sponding radial  lines  as  before.  Lines  to  represent 
the  follower  are  now  drawn  across  each  of  these 
radial  lines,  at  the  same  angle  to  them  that  the 
follower  makes  with  the  vertical  center  line  when 


168         SELF-TAUGHT  MECHANICAL  DRAWING 

at  that  part  of  its  stroke  corresponding  to  the  par- 
ticular radial  line  across  which  the  line  represent- 
ing the  follower  is  being  drawn.  A  curved  line 
passing  along  tangent  to  all  of  these  lines  gives 
the  shape  of  the  cam  as  before. 

Design  a  Cam  with  a  Round  Follower  Rising  Ver- 
tically.— In  Fig.  138  the  follower  has  the  same  uni- 
form rise,  and  the  same  periods  of  rest  as  before. 


FIG.  138.— Cam  with  Roller  Follower. 

A  cam  with  a  round  follower  is  less  limited  in  its 
capabilities  than  one  with  a  straight  follower ;  in 
the  one  here  shown  the  follower  on  its  return 
drops  below  the  position  in  which  it  is  shown. 
That  part  of  the  cam  during  which  the  conditions 
are  the  same  as  in  the  others  is  divided  off  and 


CAMS  169 

the  position  of  the  center  of  the  follower  upon  the 
radial  lines  is  obtained  in  the  same  manner  as 
before.  That  part  of  the  cam  representing  the 
return  of  the  follower  is  divided  into  such  angles 
as  desired,  and  the  distance  through  which  the  fol- 
lower is  to  drop  as  the  cam  revolves  through  each 
of  these  angles  is  marked  off  upon  the  proper 
radial  line.  A  curved  line  which  is  now  made  to 
pass  through  all  of  the  points  so  obtained  gives 
the  pitch  line  of  the  cam. 

In  drawing  such  a  cam  it  is  not  always  neces- 
sary to  fully  draw  the  working  faces.  The  pitch 
line  and  the  method  of  obtaining  it  being  shown, 
a  number  of  circles  representing  consecutive  posi- 
tions of  the  follower  may  be  drawn.  This  will 
usually  be  sufficient.  The  side  view  of  the  cam, 
which  in  a  case  like  this  would  naturally  be  made 
in  section,  will  give  opportunity  to  show  any  fur- 
ther detail  that  may  be  desired. 

Design  a  Cam  with  a  Round  Follower  Mounted  on 
a  Swinging  Arm. — Fig.  139  shows  such  a  cam,  all 
of  the  conditions  as  to  rise,  rest  and  return  of  the 
follower  being  the  same  as  in  the  cam  shown  in 
Fig.  138.  The  cam  is  divided  into  the  same  angles 
as  before,  and  the  position  of  the  follower  is  laid 
out  on  these  radial  lines  as  though  it  moved  ver- 
tically. These  positions  are  then  modified  in  the 
following  manner :  Draw  the  arc  ab  representing 
the  path  of  the  center  of  the  follower  as  it  rises, 
and  extend  the  dotted  circular  lines,  which  repre- 
sent successive  heights  of  the  follower,  from  the 
vertical  center  line  to  this  arc.  The  distance  of 
each  of  the  intersections  of  the  dotted  circular 


170         £ELF-TAUGHT  MECHANICAL  DRAWING 

lines  with  the  arc  a&,  from  the  vertical  center  line 
is  then  taken  with  the  compasses  and  is  marked 
off  upon  the  same  dotted  line  from  the  radial  line 
at  which  it  terminates,  or,  where  the  follower  has 
a  period  of  rest,  from  both  of  the  radial  lines 


FIG.  139.— Cam  with  Roller  Follower  Mounted  on 
Swinging  Arm. 

where  the  period  of  rest  takes  place.  Thus  the  dis- 
tance of  the  point  1  from  the  vertical  center  line  is 
marked  back  upon  the  dotted  circular  line  from  the 
radial  lines  ra  and  n.  Point  2  is  marked  back  from 
the  radial  line  o.  Point  3  is  marked  back  from  the 
line  p.  By  this  means  the  position  which  the  fol- 
lower will  occupy,  when  each  of  the  radial  lines 
has  become  vertical,  as  the  cam  revolves,  is  deter- 


CAMS  171 

mined.  A  curved  line  which  is  made  to  pass 
through  all  of  these  points  will  be  the  required 
pitch  line  of  the  cam.  The  method  of  getting  the 
working  face  of  the  cam  is  indicated  by  the  small 
dotted  circular  arcs,  which  are  drawn  with  a  radius 
equal  to  that  of  the  follower.  It  will  be  noticed 
that,  as  the  follower,  on  its  return,  drops  below 
the  position  in  which  it  is  shown,  it  passes  to  the 
other  side  of  the  vertical  center  line,  so  that  in 
marking  off  its  position  from  the  radial  lines  x  and 
y  this  must  be  borne  in  mind.  The  question  as  to 


FIG.  140.— Reciprocating  Motion  Cam. 

on  which  side  of  a  radial  line  the  new  position  of 
the  follower  will  be,  may  be  readily  determined  by 
imagining  the  cam  to  revolve  so  as  to  bring  that 
particular  line  vertical. 

Reciprocating  Cams. — Fig.  140  shows  a  straight 
cam,  which  by  a  reciprocating  motion  imparts  a 
sideways  motion  to  its  follower.  The  pitch  line 
of  such  a  cam  may  be  determined  by  intersecting 
lines  at  right  angles  to  each  other.  As  here  shown 
the  distance  through  which  the  follower  is  to  be 
raised  is  divided  into  a  number  of  equal  spaces  by 
horizontal  lines,  and  the  distance  through  which  it 
is  desired  to  have  the  cam  move  in  order  to  raise 
the  follower  from  one  horizontal  line  to  the  next 
one  is  indicated  by  vertical  lines.  A  curved  line 


172        SELF-TAUGHT  MECHANICAL  DRAWING 

which  is  made  to  pass  through  the  intersections  of 
these  lines  will  be  the  required  pitch  line  of  the 
cam. 

If  the  follower,  instead  of  rising  vertically,  rose 
at  ah  angle,  or  if  it  were  mounted  on  a  swinging 
arm,  the  pitch  line  would  be  modified  in  the  same 
manner  as  that  of  the  cam  shown  in  Fig.  139. 

Cams  With  a  Grooved  Edge. — It  is  sometimes  de- 
sired to  have  a  revolving  cam  impart  a  sideways 


FIG.  141.— Cam  with  Grooved  Edge. 

motion  to  a  follower.  This  is  done  by  having  a 
groove  in  the  edge  of  the  cam,  as  shown  in  Fig. 
141.  Such  a  cam  may  be  considered  as  a  modified 
form  of  a  reciprocating  cam,  and  its  pitch  line  may 
be  determined  in  the  same  way. 

By  laying  out  a  development  of  the  pitch  lino,  or 
of  that  part  of  it  which  is  to  operate  the  follower, 
as  shown  in  Fig.  142,  horizontal  lines,  that  is,  lines 
parallel  with  the  pitch  line,  may  be  drawn  to  indi- 
cate successive  stages  in  the  movement  of  the  fol- 
lower, and  lines  at  right  angles  to  these  to  indicate 


CAMS 


173 


the  desired  movement  of  the  cam.  The  pitch  line 
is  then  drawn  through  the  intersections  of  these 
lines  as  before. 

A  Double  Cam  Providing  Positive  Return. — In  a 
cam  like  that  shown  in  Fig.  138,  where  the  return 


FIG.  142. — Development  of  Cam  Action  of  Grooved-Edge 
Cam  in  Fig.  141. 

of  the  follower  is  insured  by  a  groove  in  the  face 
of  the  cam,  the  groove  must  be  slightly  broader 
than  the  diameter  of  the  cam  roller  to  insure  free- 
dom of  action,  as,  when  the  cam  is  forcing  the  rol- 


FIG.  143.— Double  Cam  Providing  Positive  Return. 

ler  away  from  the  center,  the  roller  will  revolve  in 
the  opposite  direction  to  that  in  which  it  revolves 
when  the  other  face  of  the  cam  groove  acts  on  it 
to  draw  it  toward  the  center,  so  that  unless  clear- 


174         SELF-TAUGHT  MECHANICAL  DRAWING 

ance  is  provided,  there  will  be  a  grinding  action 
between  the  roller  and  the  faces  of  the  cam  groove. 
This  clearance,  however,  causes  the  cam  to  give  a 
knock  or  blow  on  the  roller  each  time  its  action  is 
reversed,  and  the  reversal  of  the  direction  of  the 
revolution  of  the  roller  itself  causes  a  temporary 
grinding  action.  These  actions  may  become  ob- 


FlG.  144.— Positive  Return  Cam  with  Rollers  Mounted  on 
Swinging  Arms. 

jectionable,  especially  at  high  speeds.  A  method 
which  overcomes  these  objections,  and  which  is 
preferred  by  some  for  such  work,  is  shown  in  Fig. 
143,  where  the  return  is  secured  by  a  secondary 
cam  mounted  on  the  same  shaft  as  the  primary 
cam,  but  acting  on  a  roller  of  its  own.  In  this  case 
there  is  no  reversal  of  the  direction  of  the  revolu- 
tion of  the  rollers,  so  that  the  necessity  of  provid- 


CAMS  175 

ing  clearance  does  not  exist.  Where  the  forward 
and  backward  motion  of  the  rollers  is  in  a  straight 
line  passing  through  the  center  of  the  cam  shaft, 
as  in  this  case,  it  is  only  necessary  in  designing 
the  secondary  cam  to  preserve  the  distance  be- 
tween its  pitch  line  and  the  pitch  line  of  the  prim- 
ary cam  constant,  measuring  through  the  center  of 
the  cam  shaft,  as  shown  at  x  and  y. 

If,  however,  the  rollers  are  mounted  on  swing- 
ing arms,  as  shown  in  Fig.  144,  so  that  their  for- 
ward and  backward  motion  is  not  in  such  a  straight 
line,  then  the  shape  of  the  secondary  cam  will  be 
subject  to  modification  on  principles  previously 
explained.  It  is  obviously  necessary  where  this 
method  of  operation  is  used,  that  provision  be  made 
to  absolutely  prevent  any  change  in  the  relative 
position  of  the  two  cams,  as  by  bolting  them  to- 
gether, or,  better  still,  by  having  them  cast 
together  in  one  piece. 

Cams  for  High  Velocities. — In  machinery  work- 
ing at  a  high  rate  of  speed,  it  becomes  very  im- 
portant that  cams  are  so  constructed  that  sudden 
shocks  are  avoided  when  the  direction  of  motion 
of  the  follower  is  reversed.  While  at  first  thought 
it  would  seem  as  if  the  uniform  motion  cam  would 
be  the  one  best  suited  to  conditions  of  this  kind,  a 
little  consideration  will  show  that  a  cam  best  suited 
for  high  speeds  is  one  where  the  speed  at  first  is 
slow,  then  accelerated  at  a  uniform  rate  until  the 
maximum  speed  is  reached,  and  then  again  uni- 
formly retarded  until  the  rate  of  "motion  of  the 
follower  is  zero  or  nearly  zero,  when  the  reversal 
takes  place.  A  cam  constructed  along  these  lines 


176        SELF-TAUGHT  MECHANICAL  DRAWING 


FIG.  145.— Uniformly  Accelerated  Motion  Cam. 


CAMS  177 

is  called  a  uniformly  accelerated  motion  cam.  The 
distances  which  the  follower  passes  through  during 
equal  periods  of  time  increase  uniformly,  so  that, 
if,  for  instance,  the  follower  moves  a  distance  equal 
to  1  length  unit  during  the  first  second,  and  3 
during  the  second,  it  will  move  5  length  units 
during  the  third  second,  7  during  the  fourth,  and 
so  forth.  When  the  motion  is  retarded,  it  will 
move  7,  5,  3  and  1  length  units  during  successive 
seconds,  until  its  motion  becomes  zero  at  the  re- 
versal of  the  direction  of  motion  of  the  follower. 

In  Fig.  145  is  shown  a  uniformly  accelerated 
motion  plate  cam.  Only  one-half  of  the  cam  has 
been  shown  complete,  the  other  half  being  an  exact 
duplicate  of  the  half  shown,  and  constructed  in  the 
same  manner.  The  motion  of  the  follower  is  back 
and  forth  from  A  to  G,  the  rise  of  the  cam  being 
180  degrees,  or  one-half  of  a  complete  revolution. 
To  construct  this  cam,  divide  the  half-circle,  AKL, 
in  six  equal  angles,  and  draw  radii  HB± ,  HC^ , 
etc.  Then  divide  AG  first  in  two  equal  parts  AD 
and  DG,  and  then  each  of  these  parts  in  three 
divisions,  the  length  of  which  are  to  each  other  as 
1:3:5,  as  shown.  Then  with  H  as  a  center  draw 
circular  arcs  from  J5,  C,  Z>,  etc.,  to B± ,  d ,  A ,  etc. 
The  points  of  intersection  between  the  circles  and 
the  radii  are  points  on  the  cam  surface. 

If  the  half-circle  AKL  had  been  divided  into  8 
equal  parts,  instead  of  6,  then  the  line  AG  would 
have  been  divided  into  8  parts,  in  the  proportions 
1:3:5:7:7:5:3:1,  each  division  being  the  same 
amount  in  excess  of  the  previous  division  while 
the  motion  is  accelerated,  and  the  same  amount 


178        SELF-TAUGHT  MECHANICAL  DRAWING 

less  than  the  previous  division  while  the  motion  is 
being  retarded.  With  a  cam  constructed  on  this 
principle  the  follower  starts  at  A  from  a  velocity 
of  zero ;  it  reaches  its  maximum  velocity  at  D ;  and 
at  G  the  velocity  is  again  zero,  just  at  the  moment 
when  the  motion  is  reversed. 

A  graphical  illustration  of  the  shape  of  the  uni- 
formly accelerated  motion  curve  is  given  in  Fig. 


i 


BC          DE          F  GHL 


\ 


/ 


FIG.  146. — Development  and  Projection  of  Uniformly 
Accelerated  Motion  Cam  Curve. 


146.  To  the  right  is  shown  the  development  of 
the  curve  as  scribed  on  the  surface  of  a  cylindrical 
cam.  This  development  is  necessary  for  finding 
the  projection  on  the  cylindrical  surface,  as  shown 
at  the  left.  To  construct  the  curve,  divide  first 
the  base  circle  of  the  cylinder  in  a  number  of  equal 


CAMS  179 

parts,  say  12;  set  off  these  parts  along  line  AL,  as 
shown ;  only  one  division  more  than  one-half  of  the 
development  has  been  shown,  as  the  other  half  is  the 
same  as  the  first  half,  except  that  the  curve  to  be 
constructed  here  is  falling  instead  of  rising.  Now 
divide  line  AK  in  the  same  number  of  divisions  as 
the  half-circle,  the'divisions  being  in  the  proportion 
1:3:5:5:3:1.  Draw  horizontal  lines  from  the 
divisions  on  AK  and  vertical  lines  from  B,  C,  D, 
etc.  The  intersections  between  the  two  sets  of 
lines  are  points  on  the  developed  cam  curve.  These 
points  are  transferred  to  the  cylindrical  surface  at 
the  left  simply  by  being  projected  in  the  usual 
manner. 

In  order  to  show  the  difference  between  the  uni- 
form motion  cam  curve,  and  that  illustrating  the 
uniformly  accelerated  motion,  a  uniform  motion 
cylinder  cam  has  been  laid  out  in  Fig.  147.  The 
base  circle  is  here  divided  in  the  same  number  of 
equal  parts  as  the  base  circle  in  Fig.  146.  The 
divisions  are  set  off  on  line  AL  in  the  same  way. 
The  line  AK,  however,  is  divided  into  a  number  of 
equal  parts,  the  number  of  its  divisions  being  the 
same  as  the  number  of  divisions  in  the  half -circle. 
By  drawing  horizontal  lines  through  the  division 
points  on  AK,  and  vertical  lines  through  points  B, 
C,  D,  etc.,  points  on  the  uniform  motion  cam  curve 
are  found.  It  will  be  seen  that  this  curve  is  merely 
a  straight  line  AM.  The  curve  is  transferred  to  its 
projection  on  the  cylinder  surface  at  the  left,  as 
shown. 

It  is  evident  from  the  developments  of  the  two 
curves  in  Figs.  146  and  147,  that  the  uniform  motion 


180 


SELF-TAUGHT  MECHANICAL  DRAWING 


curve,  Fig.  147,  causes  the  follower  to  start  very 
abruptly,  and  to  reverse  from  full  speed  in  one 
direction  to  full  speed  in  the  opposite  direction. 
The  uniformly  accelerated  motion  curve,  Fig.  146, 
permits  the  follower  to  start  and  reverse  very 
smoothly,  as  is  clearly  shown  by  the  graphical 


A          B         C          DEFGHIL 


FIG.   147. — Development  and  Projection  of  Uniform  Motion 
Cam  Curve. 

illustration  of  the  curve.  The  abrupt  starting  and 
reversal  of  the  follower  in  the  uniform  motion 
curve  is  the  cause  why  this  form  of  cam,  while 
the  simplest  of  all  cams  to  lay  out  and  cut,  cannot 
be  used  where  the  speed  is  considerable,  without 
a  perceptible  shock  at  both  the  beginning  and  the 
end  of  the  stroke. 


CAMS 


181 


Besides  the  uniformly  accelerated  motion  cam 
curve,  quite  commonly  called  the  gravity  curve, 
on  account  of  it  being  based  on  the  same  law  of 
acceleration  as  that  due  to  gravity,  there  is  another 
curve,  the  harmonic  or  crank  curve,  which  is  quite 
often  used  in  cam  construction.  The  harmonic 
motion  curve  provides  for  a  gradual  increase  of 
speed  at  the  beginning,  and  decrease  of  speed  at  the 
end,  of  the  stroke,  and  in  this  respect  resembles 


xV 

\ 


EI.       FI        GI        H! 

FIG.  148. — Lay-out  of  Harmonic  Motion  Cam  Curve. 

the  uniformly  accelerated  motion  curve;  but  the 
acceleration,  not  being  uniform,  does  not  produce  so 
easy  working  a  cam  as  the  gravity  curve  provides 
for.  The  harmonic  motion  curve  is,  however,  very 
simple  to  lay  out,  and  for  ordinary  purposes,  where 
excessively  high  speeds  are  not  required  of  the 
mechanism,  cams  laid  out  according  to  this  curve 
are  very  satisfactory. 

The  harmonic  curve  is  laid  out  as  shown  in  Fig. 
148.     Draw  first  a  half-circle  AEL    Divide  the 


182        SELF-TAUGHT  MECHANICAL  DRAWING 

circle  in  a  certain  number  of  equal  parts.  Draw  a 
line  AI  /! ,  and  divide  this  line  in  a  number  of  equal 
parts,  the  number  of  divisions  of  Av  Jt  being  the 
same  as  that  of  the  half -circle.  Now  draw  hori- 
zontal lines  from  the  divisions  A,  B,  C,  etc.,  on  the 
half-circle,  and  vertical  lines  from  the  divisions  on 
line  AI  /! .  The  points  where  the  lines  from  corre- 
sponding division  points  intersect,  are  points  on 
the  required  harmonic  cam  curve. 

An  approximation  of  the  uniformly  accelerated 
motion  or  gravity  curve  can  be  drawn  as  shown  in 


\ 

\n 

\ 

\ 

/ 

^ 

yr 

^^ 

^^ 

Ai      Bj      c,       D,       ei      F.,       G!      Hj       i, 

FIG.  149.— Approximation  of  Uniformly  Accelerated  Motion 
Curve. 

Fig.  149.  By  using  this  approximate  method,  any 
degree  of  accuracy  can  be  attained  without  the 
necessity  of  dividing  the  vertical  line  AK,  Fig. 
146,  in  an  excessively  great  number  of  parts.  The 
approximate  curve  in  Fig.  149  is  constructed  as 
follows:  Draw  a  half-ellipse  AEI,  in  which  the 
minor  axis  is  to  the  major  axis  as  8  to  11.  Divide 
this  half-ellipse  in  any  number  of  equal  parts,  and 
divide  the  line  Ailt  in  the  same  number  of  equal 
parts.  Now  draw  horizontal  lines  from  the  division 


CAMS  183 


points  on  the  ellipse,  and  vertical  lines 
BI,  Ci,  etc.  The  points  of  intersection  between 
corresponding  horizontal  and  vertical  lines,  are 
points  on  the  cam  curve.  This  cam  curve,  as  well 
as  the  one  in  Fig.  148,  can  be  transferred  to  the 
cylindrical  surface  of  a  cylinder  cam  by  ordinary 
projection  methods,  as  shown  in  Figs.  146  and  147. 
In  Figs.  150  and  151  are  shown  two  plate  cams 
for  comparison.  The  one  in  Fig.  150  is  a  uniform 


V 

FIG.    150.— Plate   Cam   Laid      Fia.    151.— Plate   Cam   Laid 
out  for  Uniform  Motion.  out  for  Uniformly  Accel- 

erated Motion. 

motion  cam.  The  dwell  is  180  degrees,  the  rise,  90 
degrees,  and  the  fall,  90  degrees.  As  shown  by 
the  sudden  change  of  direction  of  the  cam  curve 
at  A  and  B,  there  is  considerable  shock  when  the 
follower  passes  from  its  "dwell"  to  the  " rise, "  as 
well  as  at  the  end  of  the ' '  fall. ' '  A  sudden  reversal 
takes  place  at  C,  which  also  causes  a  shock  in 
the  mechanism  connected  with  the  follower.  In  the 
uniformly  accelerated  motion  cam,  Fig.  151,  the 


184         SELF-TAUGHT  MECHANICAL  DRAWING 

passing  from  " dwell"  to  "rise, "  the  reversal  of  the 
direction  of  motion,  and  the  return  to  the  "dwell" 
position,  is  accomplished  by  means  of  smoothly 
acting  curves,  and,  even  at  high  speeds,  no  per- 
ceptible shock  will  be  noticed. 

The  examples  given  will  show  the  necessity  of 
careful  analysis  of  conditions,  before  a  certain  type 
of  cam  curve  is  selected.  In  machinery  which 
works  at  a  low  rate  of  speed,  it  is  not  important 
whether  the  follower  moves  with  a  uniform,  har- 
monic, or  uniformly  accelerated  motion ;  but  when 
the  cam  has  a  high  rotative  speed,  and  the  follower 
a  reciprocating  motion,  it  often  becomes  practically 
impossible  to  make  use  of  the  uniform  motion 
curve  in  the  cam.  In  such  cases,  as  already  men- 
tioned, the  harmonic,  or,  preferably,  the  uniformly 
accelerated  motion  curve  should  be  used  in  laying 
out  the  cam. 


CHAPTER  XI 

SPROCKET  WHEELS 

WHEN  it  is  desired  to  transmit  power  from  one 
shaft  to  another  one  quite  near  to  it,  especially  if 
the  power  to  be  transmitted  is  considerable,  so  as 
to  preclude  the  use  of  belting,  sprocket  wheels 
with  chain  are  frequently  used,  if  the  speed  is  not 
high.  Bicycles  afford  a  familiar  illustration  of 
this  sort  of  power  transmission. 

Fig.  152  shows  a  sprocket  wheel  of  a  type  similar 
to  those  used  on  bicycles  and  shows  the  method  of 
getting  the  shape  of  the  teeth.  The  chain  is  shown 
with  the  links  (on  the  side  toward  the  observer) 
removed  so  as  to  allow  of  showing  the  teeth  with- 
out dotted  lines.  The  size  of  a  sprocket  wheel  to 
fit  a  given  chain  may  be  determined  graphically  as 
follows :  A  circle,  not  shown  in  the  illustration,  is 
first  drawn  of  a  diameter  about  equal  to  that  of 
the  desired  wheel,  and  this  circle  is  spaced  off  into 
as  many  divisions  as  the  wheel  is  to  have  teeth. 
Lines  corresponding  to  the  dotted  radial  lines  in 
the  upper  half  of  the  wheel  shown,  are  drawn  from 
these  division  points  to  the  center  of  the  circle.  A 
templet,  similar  in  shape  to  that  shown  in  Fig.  154, 
is  next  cut  out  of  paper,  the  lines  ab  and  cd  being 
at  right  angles  to  each  other,  and  the  length  of  a 
link  of  the  chain,  measured  from  center  to  center 

185 


186        SELF-TAUGHT  MECHANICAL  DRAWING 

of  the  pins  as  shown  at  a,  Fig.  152,  is  marked  off 
upon  the  line  ab,  measuring  equally  each  way  from 
the  center  line  cd.  In  getting  the  length  of  the 
link  in  the  chain  it  will  be  best,  for  the  sake  of  ac- 
curacy, to  measure  off  the  length  of  a  considerable 
portion  of  the  chain,  and  with  the  spacing  com- 
passes divide  this  length  into  twice  as  many  spaces 
as  there  are  links  in  the  measured  portion  of  the 


FIG.  152.— Sprocket  Wheel  and  Chain. 

chain.  The  compasses,  being  then  set  to  exactly 
half  the  length  of  a  link,  may  be  used  to  mark  off 
the  length  of  the  link,  1 — 2,  upon  the  templet. 
Now  letting  the  angle  abc,  Fig.  155,  represent  one 
of  the  angles  into  which  the  circle  has  been  di- 
vided, bisect  it  to  get  a  center  line  bd,  and  placing 
the  templet  so  that  its  line  cd  shall  coincide  with 
this  center  line  move  it  along  until  the  points  1 — 2 
shall  coincide  with  the  lines  ab  and  cb  of  the  angle. 
These  points  being  now  marked  off  upon  the  lines, 
give  the  location  of  the  centers  of  the  pins  in  the 
chain,  and  a  line  connecting  them  will  be  one  side 


SPROCKET  WHEELS 


187 


of  the  polygon  which  forms  the  pitch  line  of  the 
wheel.  A  spiral  may  now  be  formed  upon  this 
polygon  (see  geometrical  problem  19,  Figs.  41  and 
42) ,  and  will  give  the  path  of  the  pin  as  the  chain 


FIG.  153.— Sprocket  Wheel  Designed  for  Common 
Link  Chain. 

unwinds  from  the  wheel  when  the  latter  revolves, 
as  shown  in  Fig.  152.  The  working  face  of  that 
part  of  the  tooth  in  the  wheel  lying  outside  of  the 
pitch  polygon  is  now  struck  from  such  a  center  as 
will  cause  it  to  fall  slightly  within  the  path  of  the 
chain,  as  just  obtained,  so  that  the  link  may  fall 


FIG.  154.  FIG.  155. 

FIGS.  154  and  155.— Graphical  Method  of  Laying  Out 
Sprocket  Wheel. 

freely  into  place  as  it  enters  upon  the  tooth.  Of 
course  allowance  must  be  made  all  around  for  the 
natural  roughness  of  the  casting  if  the  wheel  is  to 
be  left  unfinished.  The  length  of  the  tooth  is 
usually  made  about  equal  to  the  width  of  the  chain. 


188         SELF-TAUGHT  MECHANICAL  DRAWING 

If  a  wheel  is  to  have  many  teeth,  it  will  gener- 
ally be  accurate  enough  to  consider  the  pitch  line 
as  a  circle  of  a  circumference  equal  to  the  number 
of  the  teeth  multiplied  by  the  length  of  the  link. 
Its  diameter  will  then,  of  course,  be  found  by 
dividing  the  circumference  by  3.1416. 

In  the  case  of  the  wheel  shown  in  Fig.  152, 
should  the  pitch  line  be  regarded  as  such  a  circle 
it  would  have  a  diameter  a  little  over  a  thirty- 
second  of  an  inch  too  small,  if  the  length  of  the 
link  is  taken  at  three-quarters  of  an  inch.  If  the 
wheel  were  to  be  made  twice  as  large,  the  error 
would  be  a  little  less  than  a  sixty-fourth  of  an  inch, 
as  it  would  decrease  at  a  slightly  faster  rate  than 
that  at  which  the  number  of  the  teeth  increased.  An 
error  of  a  sixty-fourth  of  an  inch  in  the  diameter 
of  such  a  sprocket  would  be  of  but  very  little 
moment.  Where  a  sprocket  has  but  few  teeth, 
however,  it  will  be  on  the  side  of  safety  to  always 
give  to  the  pitch  line  its  true  polygonal  form,  and 
the  only  way  by  which  its  diameter  could  be  ascer- 
tained with  any  greater  accuracy  than  by  the 
method  here  given  would  be  to  calculate  it,  as  may 
be  done  by  trigonometry.  When  the  pitch  line  of 
a  sprocket  is  regarded  as  a  circle,  the  path  of  the 
chain  as  it  unwinds  will  be  regarded  as  an  involute 
(see  geometrical  problem  20). 

The  shape  of  the  rim  of  a  sprocket  wheel  will  be 
governed  by  the  style  of  the  chain  for  which  it  is 
designed.  Fig.  153  shows  a  portion  of  the  rim  of 
a  wheel  which  is  designed  for  a  common  link 
chain ;  but  whatever  the  general  shape  of  the  rim 
may  be,  the  working  faces  of  the  teeth,  or  of  the 


SPROCKET  WHEELS  189 

projections. which  correspond  to  teeth,  will  always 
be  made  on  the  principles  here  explained. 

The  speed  ratio  of  the  two  wheels  of  a  pair  of 
sprockets  will  be  inversely  as  the  number  of  teeth 
in  each.  For  instance,  if  the  large  and  the  small 
wheels  have  respectively  13  and  7  teeth,  then  the 
speed  of  the  large  wheel  will  be  to  the  speed  of 
the  small  wheel  as  7  to  13. 


CHAPTER  XII 

GENERAL  PRINCIPLES  OF  GEARING 

Friction  and  Knuckle  Gearing. — In  machinery 
it  is  frequently  necessary  to  transmit  power  from 
one  shaft  to  another  near  to  it.  For  this  purpose 
gears  are  generally  employed.  Let  a  and  6,  Fig. 
156,  be  two  such  shafts.  If  now  disks  c  and  d  are 
mounted  upon  these  shafts,  of  such  diameters  as 


FIG.  156.  -Friction  Wheels. 


FIG.  157.  -Knuckle  Gears. 


to  give  the  required  speed  ratio,  we  will  have 
gearing  in  its  simplest  form.  Such  disks,  having 
their  edges  covered  with  leather  or  other  equiva- 
lent material,  are  called  friction  gears  and  are 
sometimes  employed  on  light  work.  At  best,  how- 
ever, they  will  transmit  but  little  power. 

If  now  we  make  semi -circular  projections  at 
equal  distances  apart  upon  the  outside  of  the  cir- 
cles c  and  d,  and  cut  out  corresponding  depressions 
inside  of  the  circles,  as  shown  in  Fig.  157,  we  will 
have  a  simple  form  of  toothed  gearing  and  the  cir- 

190 


GENERAL  PRINCIPLES  OF  GEARING  191 

cles  c  and  d  will  be  the  pitch  circles.  Such  gears, 
called  knuckle  gears,  are  sometimes  employed  on 
slow-moving  work  where  no  special  accuracy  is 
required.  They  will  not  transmit  speed  uniformly. 
If  the  driver  of  such  a  pair  of  gears  rotated  at  a 
uniform  rate,  the  driven  gear  would  have  a  more 
or  less  jerky  movement  as  the  successive  teeth 
came  into  contact,  and  if  run  at  high  speed  they 
would  be  noisy.  Various  curves  may  be  employed 
to  give  to  gear  teeth  such  an  outline  that  the 
driver  of  a  pair  of  gears  will  impart  a  uniform 
speed  to  the  driven  one,  but  in  common  practice 
only  two  kinds  are  used,  the  cycloidal,  or,  as  it  is 
sometimes  called,  epicycloidal,  and  the  involute. 

Epicycloidal  Gearing. — Let  the  circles  a,  b  and 
c,  Fig.  158,  having  their  centers  on  the  same 
straight  line,  be  made  to  rotate  so  that  their  cir- 
cumferences roll  upon  each  other  without  slipping. 
If  the  circle  c  has  tracing  points  1,  2,  3  upon  its 
circumference,  and  when  we  start  to  rotate  the 
circles  point  1  is  half  way  around  from  the  posi- 
tion in  which  it  is  shown,  then  in  rotating  the  cir- 
cles sufficiently  to  bring  the  tracing  points  to  the 
position  in  which  they  are  shown,  point  1  will 
trace  the  line  1 '  inwardly  from  the  circle  a,  and  the 
line  1 "  outwardly  from  the  circle  b.  Point  2  will 
trace  the  two  lines  which  are  shown  meeting  at 
that  point,  one  inwardly  from  the  circle  a,  and  one 
outwardly  from  the  circle  6.  Point  3  will  similarly 
trace  the  two  lines  which  met  at  that  point.  Inas- 
much as  these  lines  were  traced  simultaneously  by 
points  at  a  fixed  distance  apart,  it  is  evident  that 
if  the  circle  c  were  to  be  removed,  and  the  circles 


192         SELF-TAUGHT  MECHANICAL  DRAWING 

a  and  b  were  rolled  back  upon  each  other,  these 
lines  would  work  smoothly  together,  being  in  con- 
tact and  tangent  to  each  other  at  all  times  upon 
the  line  of  the  circle  c.  If  the  circle  c  is  now  placed 
beneath  the  circle  b  in  the  position  shown,  and  the 
three  circles  are  rolled  together  as  before,  the  tra- 
cing points  would  trace  lines  inwardly  from  6,  and 


FIG.  158.— Principle   of  Epi- 
cycloidal  Gearing. 


FIG.  159.— Principle  of  Invo- 
lute Gearing. 


outwardly  from  a,  which  would  also  work  together 
smoothly  if  the  circle  c  were  removed  and  the  cir- 
cles a  and  6  were  rolled  back  upon  each  other.  It 
is  evident  that  as  the  three  circles  are  rolled 
together  the  lines  formed  by  the  tracing  points  are 
the  same  as  though  either  a  or  6  were  taken  by 
itself,  and  the  circle  c  were  rolled  either  within 
or  upon  it,  hence  the  lines  formed  by  the  tracing 
points  are  either  epicycloids  or  hypocycloids  as 
the  case  may  be,  and  so  could  be  formed  by  the 


GENERAL  PRINCIPLES  OF  GEARING  193 

plotting    method    described   in    the   geometrical 
problems. 

If  these  two  sets  of  lines  are  now  joined  together 
so  that  the  lines  which  extend  inwardly  from  a  or 
b  form  a  continuation  of  those  which  extend  out- 
wardly and  reverse  curves  are  made  at  a  distance 
from  the  first  set  equal  to  the  thickness  of  a  gear 
tooth,  and  they  are  the-n  cut  off  at  such  a  distance 
both  outside  and  inside  of  the  circles  a  and  6  as  to 
give  to  the  teeth  the  proper 
length,  it  is  evident  that 
we  will  have  a  pair  of  per- 
fectly working  gears.  The 
circles  a  and  b  would  roll 
upon  each  other  without 
slipping  and  hence  would  FlG  ^.-Definitions'  of 

be  true  pitch  circles.      The  Gear  Tooth  Terms. 

teeth  would  work  smoothly 

together  in  constant  contact,  the  point  of  contact 

being  always  on  the  line  of  the  generating  circle. 

The  length  of  the  point  of  the  gear  tooth,  that 
is  the  portion  lying  outside  of  the  pitch  line,  is 
usually  made  one-third  of  the  circular  pitch^the 
latter  being  the  distance  between  the  teeth  meas- 
ured from  center  to  center  on  the  pitch  line.  The 
distance  below  the  pitch  line  is  made  somewhat 
greater  for  the  sake  of  clearance.  For  the  names 
of  the  various  parts  of  a  gear  tooth  see  Fig.  160. 
Cast  gears  have  some  backlash  between  the  teeth 
to  allow  for  the  roughness  of  the  castings,  as 
shown  in  Figs.  161  and  163. 

It  is  evident  that  if  another  circle,  either  larger 
or  smaller,  were  substituted  for  b  in  Fig.  158,  the 


194         SELF-TAUGHT  MECHANICAL  DRAWING 


lines  formed  by  the  generating  circle  c  either 
within  or  upon  the  circle  a  would  remain  unchanged. 
Or  if  a  different  circle  were  substituted  for  a,  the 
curves  formed  within  or  upon  6  would  remain  un- 
changed. Hence  it  follows  that  all  gears  in  the 
epicycloidal  system,  having  their  teeth  formed  by 
the  same  generating  circle  and  made  of  the  same 


FIG.  162.-Rack  with 
Epicycloidal  Teeth. 


FIG.  161.— Gears  with  Epicycloidal 
Teeth. 

size,  will  work  together  correctly,  or%  as  it  is  com- 
monly expressed,  are  interchangeable. 

In  standard  interchangeable  gears  the  generat- 
ing circle  is  made  one-half  the  diameter  of  the 
smallest  gear  of  the  set,  which  has  twelve  teeth. 
This  smallest  gear  will  have  radial  flanks,  as  that 
part  of  the  working  surface  lying  within  the  pitch 
line  is  called,  because  the  hypocycloid  of  a  circle 
formed  by  a  generating  circle  of  half  its  size  will 
be  a  straight  line  passing  through  its  center. 

Fig.  161  shows  a  portion  of  a  pair  of  such  gears, 
Fig.  162  showing  the  rack. 


GENERAL  PRINCIPLES  OF  GEARING 


195 


Gears  with  Strengthened  Flanks. — A  further  ex- 
amination of  Fig.  158  will  show  that  the  curves 
formed  by  the  generating  circle  when  it  is  in  the 
upper  of  the  two  positions  in  which  it  appears, 
work  together  by  themselves,  and  those  formed 
when  it  is  in  the  lower  position  work  similarly,  so 
that  it  is  not  necessary  that  the  same  sized  gener- 


r\ 


\J 


FIG.  164.— Rack  with 
Involute  Teeth. 


FIG.  163.— Gears  with  Involute  Teeth. 

ating  circle  should  be  used  in  both  positions,  unless 
the  gears  are  to  be  members  of  an  interchangeable 
set  of  gears.  Advantage  may  be  taken  of  this  fact 
to  strengthen  the  roots  of  the  teeth  in  a  pinion. 

If,  for  instance,  in  Fig.  161,  a  smaller  generating 
circle  were  used  in  the  upper  position,  the  effect 
would  be  to  broaden  out  the  roots  of  the  teeth  in 
the  pinion,  and  to  correspondingly  round  off  the 
points  of  the  teeth  of  the  other  gear. 

Gears  with  Radial  Flanks. —Another  modification 
which  may  be  made  is  to  have  the  teeth  of  both 
gears  with  radial  flanks.  If,  for  instance,  in  Fig. 
161  a  generating  circle  were  to  be  used  in  the 


196         SELF-TAUGHT  MECHANICAL  DRAWING 

lower  portion,  of  half  the  pitch  diameter  of  the 
large  gear,  the  effect  would  be  to  give  to  that  gear 
radial  flanks,  and  to  make  the  points  of  the  teeth 
of  the  small  gear  broader  in  order  to  work  properly 
with  them.  Then  both  gears  would  have  radial 
flanks.  Such  gears  have  been  considerably  used. 
They  are  not  as  strong  as  gears  of  the  standard 
shape,  and  the  only  advantage  is  that  it  is  easier 
to  make  the  pattern,  the  teeth  being  all  worked  out 
with  a  flat-faced  plane;  but  as  the  teeth  of  in- 
volute gears,  described  in  the  next  section,  can  be 
worked  out  in  the  same  way,  and  as  such  gears  are 
interchangeable,  the  advantage  is  obviously  in 
favor  of  the  involute  system  for  such  work. 

Involute  Gears. — In  involute  gears  the  working 
surfaces  of  the  teeth  are  involutes,  formed  not 
upon  the  pitch  circles,  but  upon  base  circles  lying 
within  the  pitch  circles  and  tangent  to  a  line, 
called  the  line  of  action,  which  passes  obliquely 
through  the  point  where  the  pitch  circles  cross  the 
line  connecting  their  centers.  Let  a  and  b,  Fig.- 
159,  be  pitch  circles,  and  let  the  line  cd  be  the  line 
of  action.  Then  e  and  /,  being  made  tangent  to 
the  line  cd,  will  be  the  base  circles  upon  which 
the  involutes  are  to  be  formed.  If  now  this  line 
of  action  be  considered  as  part  of  a  thread  which 
unwinds  from  one  base  circle  and  winds  up  on  the 
other,  as  the  pitch  circles  are  revolved  back  and 
forth  upon  each  other,  then  if  tracing  points  were 
attached  to  the  thread  at  points  1,  2,  3,  4,  5  and  6, 
these  points  would  describe  involutes  outwardly 
from  the  base  circles,  which,  being  formed  simul- 
taneously in  pairs  and  each  pair  being  formed  by 


GENERAL  PRINCIPLES  OF  GEARING  197 

a  common  point,  would  work  together  smoothly 
like  those  formed  by  the  generating  circles  of  the 
epicycloidal  system.  That  the  base  circles  are  of 
such  size  as  to  just  pass  the  thread  as  the  pitch 
circles  roll  upon  each  other  is  proven  by  the  fact 
that  their  radii,  gd  and  gi,  and  he  and  hi,  the  radii 
gd  and  he  being  made  at  right  angles  to  the  line  of 
action,  are  corresponding  sides  of  similar  triangles, 
the  segments  into  which  the  line  of  action  is  di- 
vided by  the  line  of  centers  being  the  other  sides, 
and  hence  have  the  same  ratio.  It  would  only  then 


FIG.  165. -Modified  Form  of  Involute  Rack  Teeth. 

be  necessary  to  reverse  the  direction  of  the  thread 
to  get  curves  for  the  other  side  of  the  teeth,  and 
to  give  to  the  teeth  their  proper  length  inside  and 
outside  of  the  pitch  line  to  obtain  a  pair  of  cor- 
rectly working  involute  gears.  That  part  of  the 
tooth  of  an  involute  gear  which  may  lie  within  the 
base  line  is  made  radial. 

In  the  standard  interchangeable  involute  gears 
the  line  of  action  is  given  an  obliquity  of  15 
degrees  (cut  gears,  14J  degrees) .  This  angle  may 
be  readily  obtained  by  the  combination  of  the 
triangles  resting  against  the  blade  of  the  T-square 
shown  in  Fig.  166.  The  point  of  contact  of  the 


198        SELF-TAUGHT  MECHANICAL  DRAWING 


teeth  is  always  upon  the  line  of  action  and  the 
push  of  one  tooth  against  another  is  in  its  direc- 
tion, hence  its  name. 

The  teeth  of  the  15-degree  involute  rack  have 
straight  sides,  inclined  to  the  pitch  line  at  an  angle 
of  75  degrees  as  shown  in  Fig.  164.  This  shape, 
however,  is  subject  to  a  slight  modification  to  avoid 
interference  of  the  points  of  the  teeth  with  the 
radial  flanks  of  small  gears. 

Interference  in  Involute  Gears. — The  points  c 
and  d,  Fig.  159,  where  the  line  of  action  is  tangent 
to  the  base  circles,  are  called  the  limiting  points. 
If  the  involutes  which  spring  from  either  base  cir- 
cle are  so  long  as  to  reach 
beyond  these  limits  on  the 
other  base  circle,  they  will 
interfere  with  the  radial 
flanks  of  the  mating  teeth. 
At  A;  is  shown  an  elongated 
involute  interfering  with 
the  radial  flank  of  the 
mating  tooth.  This  is,  of 
course,  a  highly  exagger- 
ated case.  The  interfer- 


FIG.  166.— Obtaining  a  15- 
or  75-degree  Angle  by 
30-  and  45-degree  Tri- 
angles. 


ence  will  occur  sooner  as  the  line  of  action  is  made 
to  cross  the  line  of  centers  at  a  less  oblique  angle, 
as  in  standard  gears,  and  still  earlier  as  the  pitch 
circle  b  is  made  larger.  In  gearing  of  standard  pro- 
portions, a  gear  of  30  teeth  is  the  smallest  that  will 
work  correctly  with  a  straight  toothed  rack.  In 
the  gears  shown  in  Fig.  163,  the  teeth  of  the  large 
gear  pass  beyond  the  limiting  point  of  the  small 
gear,  and  hence,  if  made  of ,  true  involute  shape, 


GENERAL  PRINCIPLES  OF  GEARING  199 

their  extremities  will  not  work  properly  with  the 
flanks  of  the  small  gear. 

There  are  three  methods  available  to  overcome 
this  interference.  First,  to  hollow  out  the  flanks 
of  the  teeth  of  the  small  gear.  Second,  to  round 
off  the  points  of  the  teeth  of  the  large  gear.  This 
is  the  method  usually  adopted,  in  interchangeable 
gears,  the  point  being  rounded  off  enough  to  clear 
the  flanks  of  the  smallest  gear  of  the  set.  Fig.  165 
shows  the  teeth  of  the  rack  so  corrected  in  larger 
scale.  Third,  to  cut  off  that  part  of  the  tooth  in 
the  large  gear  which  extends  beyond  the  limiting 
point  of  the  small  gear.  This  is  done  in  special 
cases. 

The  Two  Systems  Compared. — The  great  point 
in  favor  of  epicycloidal  gearing  would  appear  to  be 
in  its  freedom  from  interference.  It  is  necessary, 
however,  in  order  to  have  epicycloidal  gears  run 
well,  to  have  the  pitch  circles  of  the  two  gears  of 
a  pair  just  coincide,  as  shown  in  Fig.  161;  but 
with  involute  gears  the  distance  between  centers 
may  be  varied  somewhat  without  affecting  their 
smoothness  of  operation,  though  where  the  points 
of  the  teeth  are  rounded  off  to  avoid  interference, 
as  previously  explained,  the  amount  of  variation 
which  can  be  allowed  is  not  great.  As  no  value 
has  been  given  to  the  angle  at  which  the  line  of 
action  crosses  the  line  of  centers  in  Fig.  159,  it  is 
evident  that  whether  the  base  circles  are  brought 
nearer  together  or  are  carried  further  apart,  circles 
which  might  then  be  drawn  through  the  point 
where  the  line  of  action  crosses  the  line  of  centers, 
would  roll  upon  each  other  while  the  base  circles 


200         SELF-TAUGHT  MECHANICAL  DRAWING 

passed  the  thread  as  before,  and  hence  would  be 
true  pitch  circles  for  the  time  being.  The  amount 
of  backlash,  that  is,  the  space  between  the  faces 
of  the  teeth,  would  vary,  but  the  smoothness  of 
operation  would  not  be  affected.  This  property  of 
involute  gears  is  very  valuable  in  cases  where  the 
distance  between  centers  is  variable,  as  in  rolling 
mill  gearing.  In  such  cases,  however,  interfer- 
ence must  be  avoided  by  the  first  of  the  three 
methods  explained,  that  of  hollowing  out  the  flanks 
of  the  teeth  of  the  mating  gear. 

The  epicycloidal  system  is  the  older  of  the  two, 
and  cast  gears  are  still  quite  largely  made  to  this 
system,  there  being  so  many  patterns  of  that  sys- 
tem on  hand.  But  though  the  epicycloidal  system 
once  had  the  field  to  itself,  the  fact  that  the  invol- 
ute system  has  so  largely  replaced  it,  having  al- 
most wholly  superseded  it  for  cut  gearing,  shows 
the  trend  of  modern  practice.  It  is  sometimes 
urged  against  the  involute  system  that  the  thrust 
on  the  shaft  bearings  is  greater  than  with  the  epi- 
cycloidal system,  on  account  of  the  obliquity  of  its 
line  of  action.  But  though  the  line  of  action  is  at 
an  angle  to  the  direction  of  the  motion  of  the  teeth 
when  they  are  on  the  line  connecting  their  centers, 
it  is  a  constant  angle;  while  it  is  never  less,  it  is 
never  more.  With  the  epicycloidal  system,  on  the 
other  hand,  though  the  teeth  of  the  driver  give  a 
square  push  to  the  teeth  of  the  driven  gear  when 
they  are  in  contact  on  the  line  of  centers,  yet 
the  direction  of  this  pushing  action  being  on  the 
line  of  the  generating  circle,  is  variable,  so  that 
when  the  teeth  are  first  coming  into  contact  with 


GENERAL  PRINCIPLES  OF  GEARING  201 

one  another  they  have  an  obliquity  of  action  fully 
as  great,  if  not  greater,  than  standard  involute 
gears.  For  this  reason  such  authorities  as  the 
Brown  &  Sharp  Co.,  Grant  and  Unwin,  do  not  con- 
sider this  objection  as  being  of  great  weight. 

Twenty-Degree  Involute  Gears. — It  has  been  al- 
ready shown  how  the  teeth  of  epicycloidal  gears 
may  be  considerably  strengthened  where  it  is  not 
necessary  to  have  them  interchangeable.  In  invol- 
ute gearing,  when  a  stronger  gear  is  desired  than 
the  standard  15-degree  tooth  provides  for,  recourse 
may  be  had  to  increasing  the  obliquity  of  the  line 
of  action.  This  makes  the  tooth  considerably 
broader  at  the  base,  and  correspondingly  narrower 
at  the  point.  The  angle  usually  adopted  in  such 
cases  is  20  degrees,  and  some  makers  report  an 
increasing  demand  for  such  gears. 

Shrouded  Gears.  -When  it  is  desired  to  strengthen 
the  teeth  of  cast  gears  without  increasing  their 
size,  or  without  using  any  other  than  a  standard 
shape  or  tooth,  the  practice  of  shrouding  them  is 
sometimes  resorted  to.  This  consists  in  casting 
a  flange  on  one  or  both  sides  of  the  gear.  Full 
shrouding  consists  in  having  the  flanges  extend  to 
the  points  of  the  teeth  as  shown  in  Fig.  167 ;  half 
shrouding  is  where  the  flanges  extend  only  to  the 
pitch  line  as  shown  in  Fig.  168.  When  the  two 
gears  of  a  pair  are  of  nearly  equal  size  so  that 
their  teeth  would  be  of  about  the  same  strength 
it  would  be  natural  to  use  half  shrouding  on  both 
gears  as  shown. 

When,  however,  there  is  much  difference  in  the 
size  of  the  gears,  as  shown  in  Fig.  167,  it  would  be 


202         SELF-TAUGHT  MECHANICAL  DRAWING 


natural  to  use  full  shrouding  on  the  small  gear,  as 
otherwise  its  teeth  would  be  weaker  than  those 
of  the  large  gear.  Shrouding  is  estimated  to 
strengthen  the  teeth  from  25  to  50  per  cent. 


FIG.  167. 

FIGS.  167  and  168. -Shrouded  Gears. 

Bevel  Gears. — In  cylindrical  or  spur  gears  the 
pitch  surfaces  are  cylinders  of  a  diameter  equal  to 
the  pitch  circle;  in  bevel  gears  the  pitch  surfaces 
are  cones,  having  their  apices  coinciding. 

In  designing  a  pair  of  bevel  gears  as  shown  in 
Fig.  169,  the  center  lines  ab  and  cd  are  first  drawn, 
and  the  pitch  diameters  then  laid  out  from  these 


GENERAL  PRINCIPLES  OF  GEARING 


203 


lines  as  indicated.  From  the  point  where  the  lines 
of  the  pitch  diameters  meet  at  e,  a  line  is  drawn  to 
the  point  where  the  center  lines  intersect  at  k. 
This  gives  one  side  of  the  pitch  cone  of  each  gear 
and  from  this  the  other  sides  of  the  cones  are 


FIG.  169.— Bevel  Gears. 

readily  drawn.      All  lines  of  the  working  surfaces 
of  the  gears  meet  at  the  point  h. 

To  lay  out  the  teeth,  the  line/gr  is  first  drawn 
through  the  point  e  and  at  right  angles  to  eh.  This 
gives  the  outside  face  of  the  teeth,  and  the  points 
/and  g  become  the  apices  of  cones  upon  the  devel- 
opment of  which  the  teeth  are  laid  out.  With  cen- 
ters at /and  g  the  pitch  line  developments  ei  and 
ej  are  drawn,  and  upon  these  lines  the  teeth  are 
laid  out  the  same  as  for  ordinary  gears.  When 
the  two  gears  of  a  pair  are  of  the  same  size 
they  are  called  miter  gears. 


204 


SELF-TAUGHT  MECHANICAL  DRAWING 


Worm  Gearing. — In  worm  gearing,  as  shown  in 
Fig.  170,  a  screw  having  its  threads  shaped  like 
the  teeth  of  a  rack  engages  with  the  teeth  of  a 
gear  having  a  concave  face  and  teeth  of  such  shape 
as  to  fit  the  threads  of  the  screw.  If  the  screw  is 
single  threaded,  one  rotation  of  it  will  cause  the 
gear  to  revolve  the  distance  of  one  tooth ;  if  double 
threaded,  the  gear  will  turn  two  teeth,  and  so  on. 

In  worm  gearing,  the  worm  wears  much  faster 
than  the  gear;  it  is,  therefore,  frequently  made  of 


FIG.  170.— Worm  and  Worm-Gear. 

steel  while  the  worm-wheel  is  made  of  bronze,  to 
give  the  combination  increased  durability. 

In  involute  worm  gearing  interference  is  com- 
monly avoided  by  the  last  of  the  three  methods 
already  mentioned.  The  points  of  the  thread  of  the 
screw  in  Fig.  170  project  but  little  beyond  the 
pitch  line,  the  root  spaces  of  the  gear  being  made 
correspondingly  shallow.  At  the  same  time,  the 
points  of  the  teeth  in  the  gear  are  made  long 
enough  to  preserve  their  total  length  the  same  as 
usual,  and  the  depth  of  the  screw  thread  inside  the 
pitch  Iin2  is  made  sufficient  for  clearance.  But  un- 


GENERAL  PRINCIPLES  OF  GEARING  205 

less  the  worm-gear  has  less  than  30  teeth,  the 
standard  shape  of  tooth  will  be  satisfactory. 

Circular  Pitch. — In  designing  gearing,  the  old 
method  (the  one  which  is  given  in  the  older  trea- 
tises on  the  subject)  is  to  use  the  circular  pitch; 
that  is,  the  distance  between  the  teeth,  measured 
from  center  to  center  on  the  pitch  circle.  This 
method  has  many  disadvantages.  For  instance,  if 
it  is  required  to  make  a  pattern  of  a  gear  to  mesh 
with  one  already  on  hand,  the  natural  thing  to  do 
in  measuring  up  the  old  gear  is  to  first  guess  at 
where  the  pitch  line  is,  and  then  measure  straight 
across  from  one  tooth  to  the  next.  This  leads  to 
two  errors  in  the  result;  first,  the  probably  incor- 
rect location  of  the  pitch  line,  and,  second,  the  dis- 
tance measured  is  the  chordal  pitch  instead  of  the 
circular  pitch.  A  noisy  pair  of  gears  would  quite 
likely  be  the  result. 

Again,  as  the  ratio  between  the  circumference 
and  the  diameter  of  a  circle  is  not  an  even  num- 
ber, but  a  troublesome  fraction,  the  use  of  the  cir- 
cular pitch  method  will  give  the  pitch  diameter  of 
the  gear  in  inconvenient  fractions  of  an  inch,  un- 
less an  equally  inconvenient  circular  pitch  is  used. 
This  method  has  so  many  disadvantages  that  it 
has  been  largely  replaced  by  the  more  convenient 
"diametral  pitch'*  method.  For  cut  gears  the  dia- 
metral pitch  method  is  used  almost  exclusively; 
but  for  cast  gears  there  are  so  many  patterns  on 
hand,  made  by  the  circular  pitch  method,  that  that 
method  is  still  used  considerably  on  such  work, 
especially  on  the  larger  sizes  of  gears. 

Where  one   is  designing  new  work,  however, 


206        SELF-TAUGHT  MECHANICAL  DRAWING 


where  no  old  gear  patterns  made  by  the  circular 
pitch  method  are  used,  the  diametral  pitch  method 
will  be  by  far  the  most  convenient  to  use,  which- 
ever style  of  tooth,  whether  involute  or  epicy- 
cloidal,  may  be  adopted. 

PITCH  DIAMETERS  OF  GEARS  FROM  10  TO  100 
TEETH,  OF  1-INCH  CIRCULAR   PITCH. 


No. 
of 
Teeth 

Diam.  in 
Inches 

No. 
of 
Teeth 

Diam.  in 
Inches 

No. 
of 
Teeth 

Diam.  in 
Inches 

No. 
of 
Teeth 

Diam.  in 
Inches 

10 

3.183 

33 

10.504 

56 

17.825 

79 

25.146 

11 

3.501 

34 

10.823 

57 

18.144 

80 

25.465 

12 

3.820 

35 

11.141 

58 

18.462 

81 

25.783 

13 

4.138 

36 

11.459 

59 

18.781 

82 

26.101 

14 

4.456 

37 

11.777 

60 

19.099 

83 

26.419 

15 

4.775 

38 

12.096 

61 

19.417 

84 

26.738 

16 

5.093 

39 

12.414 

62 

19.735 

85 

27.056 

17 

5.411 

40 

12.732 

63 

20.054 

86 

27.375 

18 

5.730 

41 

13.051 

64 

20.372 

87 

27.693 

19 

6.048 

42 

13.369 

65 

20.690 

88 

28.011 

20 

6.366 

43 

13.687 

66 

21.008 

89 

28.329 

21 

6.685 

44 

14.006 

67 

21.327 

90 

28.648 

22 

7.003 

45 

14.324 

68 

21.645 

91 

28.966 

23 

7.321 

46 

14.642 

69 

21.963 

92 

29.285 

24 

7.639 

47 

14.961 

70 

22.282 

93 

29.603 

25 

7.958 

48 

15.279 

71 

22.600 

94 

29.921 

26 

8.276 

49 

15.597  . 

72 

22.918  ' 

95 

30.239 

27 

8.594 

50 

15.915 

73 

23.236 

96 

30.558 

28 

8.913 

51 

16.234 

74   . 

23.555 

97 

30.876 

29 

9.231 

52 

16.552 

75 

23.873 

98 

31.194 

30 

9.549 

53 

16.870 

76 

24.192 

99 

31.512 

31 

9.868 

54 

17.189 

77 

24.510 

100 

31.831 

32 

10.186 

55 

17.507 

78 

24.828 

When  the  pitch  of  a  gear  is  given  in  inches  or 
fractions  of  an  inch,  the  circular  pitch  is  always 
meant;  as,  for  instance,  where  a  gear  is  said  to  be 
of  1-inch  pitch,  or  IJ-inch  pitch.  To  get  the  pitch 
diameter  in  such  a  case,  it  is  necessary  to  multiply 


GENERAL  PRINCIPLES  OF  GEARING  207 

this  pitch  by  the  number  of  teeth  in  the  gear,  and 
then  divide  this  product  by  3. 1416,  the  ratio  be- 
tween the  circumference  and  the  diameter.  For 
ascertaining  the  pitch  diameter  of  gears  when 
using  the  circular  pitch,  the  accompanying  table 
will  save  much  time.  If  the  gear  is  of  any  other 
than  1-inch  circular  pitch,  multiply  the  diameter 
here  given  for  the  required  number  of  teeth,  by 
the  circular  pitch  to  be  used. 

Proportions  of  Teeth. — The  proportions  of  the 
teeth  of  gears  where  the  circular  pitch  method  is 
used,  are  given  slightly  different  by  various  writ- 
ers. The  length  of  the  teeth  is  entirely  arbitrary 
and  therefore  this  discrepancy  is  quite  natural. 
It  is  also  unimportant,  excepting  as  uniformity 
is  desirable.  The  proportions  as  given  by  Grant 
are  as  follows :  The  addendum  and  dedendum  are 
each  made  one-third  of  the  circular  pitch;  the 
clearance,  the  distance  of  the  root  line  below  the 
dedendum  line,  is  made  one-eighth  of  the  adden- 
dum; the  backlash,  the  space  which  is  allowed  be- 
tween the  sides  of  the  teeth  in  cast  gears,  is  made 
about  the  same  as  the  clearance.  This  presents  the 
proportions  in  fractions  which  are  convenient  to 
use,  and  at  the  same  time  makes  the  proportions 
practically  the  same  as  those  of  the  diametral  pitch 
method.  Cut  gears  are  made  without  backlash. 

Diametral  Pitch. — In  the  diametral  pitch  method 
the  gear  is  considered  as  having  a  given  number 
of  teeth  for  each  inch  of  pitch  diameter.  Gears 
having  three,  four,  or  five  teeth  to  each  inch  of 
their  pitch  diameters  are  said  to  be  of  three,  four, 
or  five  pitch.  With  this  method  the  addendum 


208         SELF-TAUGHT  MECHANICAL  DRAWING 

(the  distance  which  the  teeth  project  beyond  the 
pitch  line)  is  made  equal  to  one  divided  by  the 
pitch,  so  that  the  addendum  on  gears  of  three,  four 
or  five  pitch  would  be,  respectively,  one-third,  one- 
fourth  or  one-fifth  of  an  inch.  The  advantages  of 
this  method  are  numerous. 

To  get  the  diametral  pitch  of  a  gear  it  is  only 
necessary  to  divide  the  number  of  teeth  by  the 
pitch  diameter,  or  to  divide  the  number  of  teeth 
plus  two,  by  the  outside  diameter.  A  complete 
set  of  rules,  as  well  as  formulas  and  examples  for 
calculating  spur  gear  dimensions,  will  be  given  in 
the  next  chapter. 

It  is  quite  a  common  practice  in  figuring  gears 
made  by  diametral  pitch  to  give  only  the  pitch  and 
the  number  of  teeth,  as  4  pitch,  18  teeth,  or  4  D. 
P.,  18  T.  The  letters  D.  P.  stand  for  diametral 
pitch,  the  letters  P.  D.  standing  for  pitch  diameter. 
The  pitch  diameter  is  then  found  by  dividing  the 
number  of  teeth  by  the  diametral  pitch.  When 
this  method  is  used,  the  circular  pitch  becomes 
of  secondary  importance,  but  may  be  found  by  di- 
viding 3.1416  by  the  diametral  pitch.  When  the 
circular  pitch  is  given  and  the  diametral  pitch  is 
desired,  divide  3.1416  by  the  circular  pitch.  The 
diameter  of  a  gear,  unless  otherwise  specified,  is 
always  understood  to  be  the  pitch  diameter.  With 
the  diametral  pitch  method,  the  pitch  diameter, 
unless  in  even  inches,  will  be  in  fractions  of  an 
inch  corresponding  to  the  pitch,  so  that  the  frac- 
tional parts  of  the  diameter  of  gears  of  three,  four 
or  five  pitch,  for  instance,  would  be  thirds,  fourths 
or  fifths  of  an  inch. 


GENERAL  PRINCIPLES  OF  GEARING  209 

The  Hunting  Tooth. — It  is  a  common  practice  in 
making  gear  patterns  to  have  the  teeth  of  the  two 
gears  of  a  pair  of  such  numbers  that  they  do  not 
have  a  common  divisor.  For  instance,  instead  of 
having  25  and  35  teeth  in  the  gears  of  a  pair,  one 
may  give  to  one  of  them  one  more  or  one  less 
tooth,  so  as  to  insure  all  of  the  teeth  of  one  gear 
coming  into  contact  with  all  of  the  teeth  of  the 
other  as  they  run  together. 

This  practice  is  condemned  by  some,  however, 
on  the  ground  that  if  any  of  the  teeth  are  of  bad 
shape  it  would  be  better  to  confine  their  injurious 
action  within  as  narrow  limits  as  possible,  rather 
than  to  have  them  ruin  all  of  the  teeth  of  the  other 
gear ;  but  the  shape  of  badly  formed  teeth  should 
be  corrected  as  soon  as  the  error  is  discovered. 

Approximate  Shapes  for  Cycloidal  Gear  Teeth.— 
That  part  of  the~cycloidal  curve  which  is  used  in 
the  formation  of  gear  tooth  outlines  is  so  short 
that  it  may  be  replaced  with  a  circular  arc  which 
will  very  closely  approximate  it,  and  such  arcs  are 
generally  used  in  the  practical  construction  of  gear 
patterns.  In  the  following  is  given  a  table  of  such 
arcs  with  the  location  of  the  centers  from  which 
they  are  struck.  The  center  from  which  that  part 
of  the  tooth  lying  outside  of  the  pitch  line  is 
drawn,  the  face  of  the  tooth,  will  be  inside  of  the 
pitch  line,  while  the  center  from  which  that  part 
of  the  tooth  lying  inside  of  the  pitch  line  is  drawn, 
the  flank  of  the  tooth,  will  be  outside  of  the  pitch 
line.  These  radii  and  center  locations  were  ob- 
tained directly  from  a  set  of  tooth  outlines  of 
3-inch  circular  pitch,  formed  by  rolling  a  genera- 


210 


SELF-TAUGHT  MECHANICAL  DRAWING 


ting  circle,  drawn  upon  tracing  paper,  upon  a  set 
of  pitch  circles,  correct  rotation  being  assured  by 
the  use  of  needle  points  pricked  through  the  gen- 
erating circle  into  the  pitch  circle,  the  needle 
points  serving  as  pivots  upon  which  the  genera- 
ting circle  was  swung  through  short  successive 
stages,  the  forward  movements  of  the  tracing 
point  in  forming  the  cycloidal  curves  being  also 
pricked  through.  Needle  points  were  also  used  in 
the  instruments  which  were  used  for  tracing  this 
curve  when  the  radius  and  center  location  were 
determined. 

CYCLOIDAL   TOOTH   OUTLINES 

Radii  and  center  locations  for  one-inch  circular  pitch.  For 
any  other  pitch  multiply  the  given  figure  by  the  required 
pitch. 


Number  of 
Teeth. 

Face 
Radius. 

Inside  of 
Pitch  Line. 

Flank 
Radius. 

Outside  of 
Pitch  Line. 

12 

0.625  ins. 

0.016  ins. 

Radial 

14 

0.666 

0.021 

4.00  ins. 

2.35    ins. 

16 

0.697 

0.026 

2.80 

1.33 

18 

0.724 

0.031 

2.37 

_ 

0.96 

20 

0.750 

0.036 

2.14 

0.73 

25 

0.802 

0.042 

1.91 

0.58 

30 

0.844 

0.052 

1.79 

0.48 

40 

0.906 

0.062 

1.64 

0.375 

60 

0.958 

0.083 

1.50 

0.29 

100 

1.010 

0.095 

1.33 

0.21 

200 

1.040 

0.120 

1.23 

0.177 

Rack 

1.080 

0.127 

1.08 

0.127 

If  the  diametral  pitch  method  is  being  used,  the  corresponding  circular 
pitch  may  be  found  by  dividing  3.1416  by  the  diametral  pitch,  as  already 
mentioned. 

Involute  Teeth.— The  construction  of  a  correct 
involute  tooth  outline  is  so  simple  a  matter  as  to 
make  the  use  of  tables  of  approximate  circular 


GENERAL  PRINCIPLES  OF  GEARING  211 

arcs  .unnecessary.  An  involute  may  be  formed  by 
the  plotting  method  given  in  the  geometrical  prob- 
lems, but  in  most  cases  it  may  be  more  readily 
formed  by  the  use  of  a  sharply  pointed  pencil 
guided  by  a  strong  thread  as  shown  in  Fig.  171, 
where  ab  represents  the  pitch  line  of  a  gear,  and 
cd  represents  the  base  circle,  having  a  number  of 
pins  stuck  into  it  at  short  distances  apart.  The 
thread  being  doubled,  forms  a  loop  to  hold  the  pen- 
cil point.  The  thread  being  drawn  tightly  around 
the  pins,  the  pencil  is  swung  outward  from  the 


FIG.  171. — Laying  out  an  Involute  Gear  Tooth. 

base  circle,  forming  the  required  involute.  When 
gears  of  over  thirty  teeth  are  to  mesh  into  others 
of  less  than  that  number,  it  will  be  necessary  to 
slightly  round  over  the  points  of  the  teeth  to  avoid 
interference  with  the  radial  flanks  of  the  mating 
gear.  For  this  purpose  use  a  radius  of  2.10  inches 
divided  by  the  diametral  pitch,  with  a  center  on 
the  pitch  line  as  shown  in  Fig.  172.  This  radius, 
2.10  inches  divided  by  the  diametral  pitch,  is  the 
same  as  that  given  by  Grant  for  rounding  off  the 
points  of  the  teeth  of  racks ;  but  actual  trial  on 
teeth  of  large  size  shows  it  to  be  correct  for  gear 
wheels  also,  giving  a  curve  which  coincides  very 
closely  with  the  epicycloidal  shape  which  the  point 


212         SELF-TAUGHT  MECHANICAL  DRAWING 

should  have  to  work  correctly  with  the  radial  flank 
of  the  mating  gear. 

That  part  of  an  involute  tooth  lying  within  the 
base  circle  is  made  radial,  as  previously  stated,  and 
a  good  fillet  should  be  drawn  in  at  the  root.  For 
this  purpose  use  a  radius  of  one-twelfth  of  the 
circular  pitch.  A  templet  which  is  fitted  to  this 


FlG.  172.— Modified  Tooth  Form  to  Avoid  Interference. 

outline  is  used  to  finish  the  drawing,  and  to  mark 
out  the  teeth  on  the  pattern. 

On  large  work  the  size  of  the  base  circle  may  be 
obtained  by  calculation  more  readily  than  by  the 
use  of  the  triangle,  as  shown  in  Fig.  166.  When 
the  line  of  action  has  an  obliquity  of  15  degrees, 
the  diameter  of  the  base  circle  will  be  equal  to 
0.966  of  the  pitch  diameter.  For  20-degree  invo- 
lute gears  the  diameter  of  the  base  circle  will  be 
0.94  of  the  pitch  diameter. 

With  the  20-degree  involute  system  the  teeth  of 
the  rack  have  an  inclination  of  70  degrees  to  the 
pitch  line.  With  this  system  there  will  be  no  ne- 
cessity for  rounding  off  the  points  of  the  teeth  of 
the  rack  or  of  a  large  gear  unless  it  meshes  with 


GENERAL  PRINCIPLES  OF  GEARING  213 

a  gear  of  less  than  18  teeth.  When,  to  avoid  inter- 
ference, it  does  become  necessary  to  round  off  the 
points  of  the  teeth  of  the  rack  or  of  large  gears, 
the  same  radius,  2.10  inches  divided  by  the  diam- 
etral pitch,  is  to  be  used,  as  in  the  15-degree 
system,  the  center  being  on  the  pitch  line  as 
before. 

Proportions  of  Gears. — A  somewhat  common  rule 
is  to  make  the  rim  and  the  arms  of  about  the  same 
thickness  as  the  teeth  at  the  root,  though  some 
make  the  thickness  of  the  rim  equal  to  the  height 
of  the  tooth ;  and  to  make  the  diameter  and  length 
of  the  hub  about  equal  to  about  twice  the  diameter 
of  the  shaft.  On  spoked  gears,  the  rim  is  also  stiff- 
ened by  ribbing  it  between  the  arms.  On  a  light 
gear  mounted  on  a  relatively  large  shaft  it  would 
be  natural  to  lighten  the  hub  somewhat.  The 
width  of  tthe  face  of  cast  gears  is  usually  made 
from  two  to  three  times  the  circular  pitch.  The 
face  of  bevel  gears  should  not  exceed  one-fifth  of 
the  diameter  of  the  large  gear,  and  the  face  of 
worm  gears  should  not  exceed  one-half  of  the 
diameter  of  the  worm. 

Strength  of  Gear  Teeth. — When  a  gear  is  to  be 
designed  for  a  given  work,  the  first  question  is 
how  large  to  make  the  teeth  to  give  the  required 
strength.  On  their  size  will  also  depend  the  gen- 
eral proportions  of  the  gear. 

It  is  comparatively  easy  to  determine  the  work 
which  the  teeth  are  doing,  that  is,  the  strain  or 
load  which  they  are  bearing,  when  the  power 
which  the  gear  transmits  is  known.  A  horse-power 
being  the  power  required  to  lift  33,000  pounds  one 


214         SELF-TAUGHT  MECHANICAL  DRAWING 

foot  in  one  minute,  the  load  on  the  teeth  will  be 
33,000  multiplied  by  the  horse-power  which  is 
being  transmitted,  and  divided  by  the  velocity  of 
the  pitch  line  of  the  gear  in  feet  per  minute ;  or, 
what  is  the  same  thing,  126,050  multiplied  by  the 
horse-power,  and  divided  by  the  product  of  the 
pitch  diameter  in  inches  multiplied  by  the  number 
of  revolutions  per  minute.  This  latter  figure, 
126,050,  takes  into  account  the  fact  that  in  the  first 
case  the  velocity  is  expressed  in  feet,  while  in  this 
case  the  diameter  is  in  inches,  and  also  the  fact 
that  the  velocity  is  a  factor  of  the  circumference 
instead  of  the  diameter. 

While  the  load  on  the  teeth  may  be  readily 
determined,  the  question  of  how  large  they  should 
be  made  to  bear  it  is  one  where  authorities  have 
differed  very  much  on  account  of  the  number  of 
factors  involved.  First  of  all  is  the  question  of 
the  material,  usually  cast  iron,  which  is  a  variable 
quantity,  both  on  account  of  the  nature  of  the 
material  itself,  different  grades  varying  greatly  as 
to  strength,  and  the  liability  of  defects  in  the  cast- 
ing. Then  there  is  the  question  of  whether  the 
load  should  be  considered  as  divided  between  two 
or  more  teeth  or  carried  by  one  tooth,  or  the  cor- 
ner of  a  tooth. 

Then  there  is  the  nature  of  the  work:  whether, 
the  load  will  be  uniform  or  whether  the  teeth  will 
be  subject  to  severe  strain  or  shock.  There  are 
questions  of  the  shape  of  the  tooth,  and  the  velocity 
at  which  the  gear  is  running,  the  teeth  having 
greater  strength  at  slow  speeds  than  at  high  speeds 
due  to  the  shocks  accompanying  high  velocities. 


GENERAL  PRINCIPLES  OF  GEARING  215 

To  show  the  different  results  given  by  different 
writers  we  may  take  the  case  of  a  gear  24  inches 
diameter,  2  inches  circular  pitch,  4  inches  face, 
running  at  100  revolutions  per  minute.  A  rule 
given  by  Box  in  his  treatise  on  mill  gearing,  and 
quoted  by  Grant  and  Kent,  would  make  the  gear 
safe  for  9.4  horse-power.  The  rule  in  Nystrom's 
Mechanics  gives  12.2  horse-power.  Rules  by  other 
writers,  quoted  by  Kent,  give  results  as  follows: 
Halsey,  22.6;  Jones  &  Laughlin,  35;  Harkness,  38; 
Lewis,  65.2.  The  rule  by  Prof .  Harkness  is  the 
result  of  investigations  conducted  by  him  in  1886. 
He  examined  a  great  many  rules,  largely,  how- 
ever, for  common  cast  gears.  Mr.  Lewis's  method, 
the  result  of  his  investigations  of  modern  machine 
molded  and  cut  gears,  though  giving  much  higher 
results  than  the  others,  is  said  to  have  proved  sat- 
isfactory in  an  extensive  practice,  and  so  may  be 
considered  reliable  for  gears  which  are  so  well 
made  that  the  pressure  bears  along  the  face  of  the 
teeth  instead  of  upon  the  corners. 

It  is  customary  in  calculating  gears  to  proceed 
on  the  assumption  that  the  load  is  borne  by  one 
tooth,  and  in  ordinary  work,  the  size  of  the  tooth 
may  be  determined  by  the  load  it  may  safely  bear 
per  inch  of  face  and  per  inch  of  circular  pitch. 

In  1879,  J.  H.  Cooper  selected  an  old  English  rule 
giving  the  breaking  load  of  the  tooth  as  2000  X 
pitch  X  face,  which,  allowing  a  factor  of  safety  of 
10,  would  give  us  a  safe  load  of  200  X  pitch  X  face. 
Kent  says  of  this  rule  that  for  rough  ordinary  work 
it  "is  probably  as  good  as  any,  except  that  the  fig- 
ure 200  may  be  too  high  for  weak  forms  of  tooth, 


216         SELF-TAUGHT  MECHANICAL  DRAWING 

and  for  high  speeds.  "  Lewis  also  considers  this 
rule  as  a  passably  correct  expression  of  good  gen- 
eral averages. 

The  value  given  by  Nystrom  and  those  given  by 
Box  for  teeth  of  small  pitch,  are  so  much  smaller 
than  those  of  other  authorities  that  Kent  says  they 
may  be  rejected  as  giving  unnecessary  strength. 
Accepting  the  factor  200  as  a  good  average  would 
leave  one  room  for  the  exercise  of  individual  judg- 
ment for  the  particular  case  in  hand.  If  the  speed 
were  slow  and  the  teeth  were  of  strong  shape,  as 
where  both  the  gears  of  a  pair,  or  all  of  the  gears 
of  a  train,  have  a  reasonably  large  number  of 
teeth,  a  higher  figure,  perhaps  225  or  more,  might 
be  taken;  while  if  the  speed  were  higher  and  one 
of  the  gears  had  but  few  teeth,  giving  them  a 
weak  form,  or  if  they  were  to  be  subject  to  much 
vibration  or  shock,  a  lower  figure,  perhaps  as  low 
as  125,  might  be  taken. 

To  ascertain  the  horse-power  safely  transmitted 
by  an  existing  gear,  we  would  then  multiply  to- 
gether its  diameter,  pitch  (circular)  and  face, 
taken  in  inches,  and  the  number  of  revolutions 
per  minute,  and  multiply  their  product  by  200,  or 
whatever  figure  is  selected,  and  divide  the  total 
product  by  126,050.  This  may,  perhaps,  be  ex- 
pressed clearer,  as  follows: 

diam.  X  rev.  X  circ.  pitch  X  face  X  200 
Horse-power=  " 


The  figure  200  would  give  to  the  24-  inch  gear 
previously  considered  30.5  horse-power.  The  fig- 
ure 125  would  give  19.0  horse-power. 


GENERAL  PRINCIPLES  OF  GEARING  217 

To  ascertain  the  size  of  the  teeth  to  transmit  a 
given  horse-power  we  may  transpose  the  above  rule 
and  say  that  the  product  of  the  pitch  multiplied 
by  the  face  would  be  equal  to  126,050  multiplied 
by  the  horse-power,  and  divided  by  the  product  of 
the  diameter  in  inches,  the  number  of  revolutions 
per  minute,  and  200,  or  the  figure  selected;  that  is: 

~.          ,  ,      ,  ,  126^050  X  horse-power 

Circ.  pitch  X  face  :=-- 


Assuming  some  pitch  and  dividing  this  result 
by  it  would  give  the  breadth  of  face.  A  few 
trials  will  give  the  desired  ratio  between  pitch 
and  breadth  of  face.  If  one  has  a  table  of  square 
roots  at  hand,  the  work  may  be  simplified  by 
assuming  some  desired  ratio,  when  the  pitch  will 
be  the  square  root  of  the  quotient  of  this  figure, 
pitch  multiplied  by  the  face,  divided  by  the  ratio. 
If,  for  instance,  the  pitch  multiplied  by  the  face 
were  found  to  be  12,  and  we  desired  them  to  be 
in  the  ratio  of  2J  to  1,  the  pitch  would  be  equal  to 
the  square  root  of  the  quotient  of  12  divided  by 
2i,  or  2.  191,  which  would  be  about  the  same  as  li 
diametral  pitch. 

Example.  —  Required  the  size  of  the  teeth  of  a 
gear  18  inches  in  diameter,  to  run  120  revolutions 
per  minute,  which  shall  transmit  five  horse-power, 
allowing  200  pounds  load  per  inch  of  face,  and 
inch  of  pitch.  Then  : 

126,050  X  5        630,250 
Pitch  X  face  =-  =  "432^000  =  L46 


nearly.     A  circular  pitch  of  0.785  inch,  correspond- 


218         SELF-TAUGHT  MECHANICAL  DRAWING 

ing  to  4  diametral  pitch,  would  give  a  breadth  of 
face  of  about  15  inches.  For  bevel  gears  take  the 
diameter  and  pitch  at  the  middle  of  the  face. 

Mr.  Lewis's  method  differs  from  the  preceding 
in  that  instead  of  using  a  single  constant,  as  200 
pounds  per  inch  of  pitch  and  inch  of  face,  two 
constants  are  used,  one,  Y,  a  factor  of  strength 
depending  on  the  number  of  teeth  in  the  gear,  and 
another,  S,  a  safe  working  stress  for  different 
speeds  of  the  pitch  line,  in  feet  per  minute.  The 
values  of  these  constants  are  given  in  the  accom- 
panying tables. 

The  rule  to  get  the  horse-power  of  a  given  gear 
is: 

TJ  p  =  circ.  pitch  X  face  X  velocity  X  S  X  Y 

33,000 

the  velocity  being  that  at  the  pitch  line  in  feet  per 
minute,  and  the  values  of  S  and  Y  being  taken 
from  the  tables.  The  velocity  is,  of  course,  the 
diameter  in  feet  X  3.1416  X  number  of  revolu- 
tions. If  the  diameter  were  taken  in  inches  then 
the  total  product  would  be  divided  'by  12.  The 
product  of  the  pitch  multiplied  by  the  face,  to 
determine  the  size  of  teeth  to  transmit  a  given 
power,  would  then  be 

33,000  X  H.  P. 

Circ.  pitch  X  face  =  — v    .T— ~ "0 ~-  ^ 
velocity  X  S  X  Y. 

The  calculation  should  be  made  for  the  gear  of 
the  pair  or  train  having  the  fewest  teeth,  as  it 
would  be  the  weakest,  unless  it  were  made  of  some 
stronger  material  as  steel,  or  unless  it  were 


GENERAL  PRINCIPLES  OF  GEARING 


219 


WORKING  STRESS,  S,   FOR  DIFFERENT  SPEEDS 

AT  PITCH  LINE  IN  FEET  PER  MINUTE, 

FOR  CAST  IRON. 


Speed. 

s. 

Speed. 

s. 

100  or  less 
200 
300 
600 

8000 
6000 
4800 
4000 

900 

1200 
1800 
2400 

3000 
2400 
2000 
1700 

shrouded.     If  made  of  steel  S  might  be  taken  2i 
times  the  tabulated  values. 

As  a  gear  with  cut  teeth  has  from  two  to  three 
times  the  strength  of  one  with  cast  teeth,  because 
of  the  more  perfect  contact,  Mr.  Lewis's  method 
might  be  adapted  to  common  cast  gears  by  taking 
the  value  of  S  at  from  one-half  to  one-third  of  the 
tabulated  value.  By  so  doing  one  could  bring  into 
the  calculation  the  question  of  shape  of  teeth  and 

FACTOR   FOR   STRENGTH,  Y,  TO   BE   USED    IN 
LEWIS'S   FORMULAS. 


0.078 
0.083 
0.088 
0.092 
0.094 
0.096 
0.098 
0.100 


13    3 
0.3 -S-i 


0.067 
0.070 
0.072 
0.075 
0.077 
0.080 
0.083 
0.087 


0.102 
0.104 
0.106 
0.108 
0.111 
0.114 
0.118 
0.122 


0.090 
0.092 
0.094 
0.097 
0.100 
0.102 
0.104 
0.107 


43 

50 

60 

75 

100 

150 

300 

Rack 


0.126 
0130 
0.134 
0.138 
0.142 
0.146 
0.150 
0.154 


0.110 
0.112 
0.114 
0.116 
0.118 
0.120 
0.122 
0.124 


220        SELF-TAUGHT  MECHANICAL  DRAWING 

speed,  which  would  be  especially  desirable  if  the 
speed  were  high  or  the  teeth  of  weak  form.  Tak- 
ing S  at  one-half  the  tabulated  value  would  give 
to  the  24-inch  gear  previously  considered  about 
the  same  power  as  allowing  200  pounds  per  inch 
of  pitch  and  face,  which  Mr.  Lewis  considers  a 
fair  value.  With  cast  gears  where  interchange- 
ability  is  not  a  necessary  feature,  the  teeth  of  a 
small  gear  could  of  course  be  considerably  strength- 
ened in  the  manner  previously  indicated  for  epicy- 
cloidal  gears;  or  the  20-degree  system  might  be 
used  if  the  teeth  have  the  involute  form. 

Thurston's  Rule  for  Shafts.— The  size  of  shaft 
which  the  gear  will  require  may  be  found  by  the 
rule  given  by  Thurston.  Multiply  the  horse-power 
to  be  transmitted  by  125  for  iron,  or  by  75  for  cold 
rolled  iron,  and  divide  the  product  by  the  number 
of  revolutions  per  minute.  The  cube  root  of  the 
quotient  will  be  the  size  of  the  shaft. 
.  The  size  of  gear  to  give  a  required  speed  may 
be  readily  determined  from  the  fact  that  the  prod- 
uct of  the  speed  of  the  driving  shaft  multiplied  by 
the  size  of  the  driving  gear  or  gears,  should  be 
equal  to  the  product  of  the  speed  of  the  driven 
shaft,  multiplied  by  the  size  of  the  driven  gear  or 
gears.  This,  perhaps,  may  be  made  clearer  by 
placing  the  driving  members  on  one  side  of  a  line, 
and  the  driven  members  on  the  other  side,  as  in 
the  following  example. 

A  shaft  making  75  turns  per  minute  has  on  it  a 
gear  of  200  teeth.  Required  the  size  of  gear  to 
mesh  with  it  which  shall  drive  its  shaft  120 


GENERAL  PRINCIPLES  OF  GEARING  221 

revolutions  per  minute.     Letting  x  represent  the 
size  of  the  required  gear  we  have 


Rev.  driving  shaft        =    75 
Size  driving  gear         =  200 


x      =  size  driven  gear. 
120  =  rev.  driven  shaft. 


Then  as  the  product  of  the  numbers  on  one  side 
of  the  line  equals  the  product  of  those  on  the  other 
side,  75  X  200  -5-  120  will  give  the  value  of  x,  the 
number  of  teeth  in  the  driven  gear.  This  method 
applies  to  a  train  of  gears  as  well  as  a  pair. 


CHAPTER  XIII 

CALCULATING  THE  DIMENSIONS  OF  GEARS 

IN  the  previous  chapter,  the  general  principles 
of  gearing  have  been  explained.  The  three  kinds 
of  gearing  most  commonly  in  use,  spur  gearing, 
bevel  gearing  and  worm  gearing,  have  been 
touched  upon,  and  the  fundamental  rules  for  the 
dimensions  of  gear  teeth  have  been  given.  In 
this  chapter  it  is  proposed  to  give  in  detail  the 
rules  and  formulas  for  these  three  classes  of  gears, 
so  as  to  enable  the  student  to  calculate  for  himself 
any  general  problem  in  gearing  with  which  he 
may  meet. 

Spur  Gearing.  —  In  the  following,  machine  cut 
gearing  is,  in  particular,  referred  to;  but  the  gen- 
eral formulas  are,  of  course,  of  equal  value  for  use 
when  calculating  cast  gears.  The  expressions  pitch 
diameter,  diametral  pitch  and  circular  pitch  have 
already  been  explained,  and  rules  have  been  given 
for  transferring  circular  pitch  into  diametral 
pitch,  and  vice  versa.  These  rules,  expressed  as 
formulas,  would  be: 


in  which  P  =  diametral  pitch,  and 
P'=  circular  pitch. 

Assume  as  an  example  that  the  diametral  pitch 
222 


CALCULATING  THE  DIMENSIONS  OF  GEARS    223 

of  a  gear  is  4.    What  would  be  the  circular  pitch 
of  this  gear? 
Using  the  formula  given,  we  have: 


pf  =  ==  0.7854  inch. 

When  the  diametral  pitch  and  the  pitch  diameter 
are  known,  the  .number  of  teeth  may  be  found  by 
multiplying  the  pitch  diameter  by  the  diametral 
pitch,  as  already  mentioned  in  the  previous  chap- 
ter. This  rule,  expressed  as  a  formula,  would  be  : 
N=PXD 

in  which  N  =  number  of  teeth, 

D  =  pitch  diameter,  and 
P  =  diametral  pitch. 

Assume  that  the  diametral  pitch  of  a  gear  is  4 
and  the  pitch  diameter  6i  inches.  What  would  be 
the  number  of  teeth  in  this  gear? 

By  inserting  the  given  values  in  the  formula 
above,  we  would  have  : 

N  =  4  X  6i  =  25  teeth. 

If  the  number  of  teeth  and  pitch  diameter  of  the 
gear  are  known,  and  the  diametral  pitch  is  to  be 
found,  a  rule  and  formula  for  this  may  be  arrived 
at  by  merely  transposing  the  rule  and  formula  just 
given.  The  diametral  pitch  equals  the  number  of 
teeth  divided  by  the  pitch  diameter,  or,  expressed 
as  a  formula: 


in  which  P,  N  and  D  signify  the  same  quantities 
as  in  the  previous  formula. 


224         SELF-TAUGHT  MECHANICAL  DRAWING 

Assume,  for  an  example,  that  the  number  of 
teeth  in  a  gear  equals  35  and  that  the  pitch  diam- 
eter is  3J  inches.  What  is  the  diametral  pitch? 

If  we  insert  the  known  values  in  the  given  for- 
mula, we  have  : 

35 

P  =  ^f  =  10  diametral  pitch. 
05 

Finally,  if  the  diametral  pitch  and  the  number 
of  teeth  are  known,  the  pitch  diameter  is  found 
by  dividing  the  number  of  teeth  by  the  diametral 
pitch,  which  rule  expressed  as  a  formula,  would  be: 


As  an  example,  assume  that  the  number  of  teeth 
in  a  gear  is  58  and  the  diametral  pitch  6.  What  is 
the  pitch  diameter  of  this  gear? 

By  inserting  the  known  values  in  the  formula, 
we  find  : 

D  =  5|-  =9.667  inches> 

If  it  now  be  required  to  find  the  outside  diam- 
eter of  the  gear,  that  is,  the  diameter  of  the  gear 
blank,  we  make  use  of  the  following  rule  :  The 
outside  diameter  equals  the  number  of  teeth  plus 
2,  divided  by  the  diametral  pitch.  Expressed  as 
a  formula,  this  rule  is: 

TV          N+2 

P 

in  which  D  '  =  outside  diameter  of  gear,  and  N 
and  P  have  the  same  significance  as  before. 
As  an  example,  assume  that  the  number  of  teeth 


CALCULATING  THE  DIMENSIONS  OF  GEARS    225 

is  58  and  the  diametral  pitch  6.  By  inserting  these 
values  in  the  formula,  we  find  the  outside  diameter: 

rv     58+2       60 
D'  =  —  -  —  =  -^-=  10  inches. 
b  b 

When  the  pitch  diameter  and  the  diametral  pitch 
are  known,  the  outside  diameter  is  found  as 
follows:  Add  the  quotient  of  2  divided  by  the 
diametral  pitch  to  the  pitch  diameter;  the  sum  is 
the  outside  diameter.  This  rule,  expressed  as  a 
formula,  is: 


in  which  the  letters  have  the  same  significance  as 
before. 

Assume  that  the  pitch  diameter  of  a  gear  is  9.667 
inches,  and  the  diametral  pitch  6.  Find  the  out- 
side diameter. 

By  inserting  the  given  values  in  the  formula,  we 
have: 

Df=  9.667  +  ~    =  9.667  +  0.333  =  10  inches. 
o 

By  a  transposition  of  the  rule  and  formula  just 
given,  we  find  that  the  pitch  diameter  equals  the 
outside  diameter  minus  the  quotient  of  2  divided 
by  the  diametral  pitch.  This  rule,  written  as  a 
formula,  is- 

D  =  u-  Jr 

Assume  that  the  diametral  pitch  of  a  gear  is  8, 
and  the  outside  diameter  12  inches.  What  is  the 
pitch  diameter? 

D  =  12-~f-=12-i  =  111  inches. 

o 


226         SELF-TAUGHT  MECHANICAL  DRAWING 

When  the  number  of  teeth  and  outside  diameter 
are  known,  the  diametral  pitch  may  be  found  by 
adding  2  to  the  number  of  teeth  and  dividing  the 
sum  by  the  outside  diameter;  or,  expressed  as  a 
formula: 

N  +  2 


P  = 


D'. 


If  the  number  of  teeth  in  a  gear  is  96  and  the 
outside  diameter  is  14  inches,  what  is  the  diame- 
tral pitch? 

If  the  known  values  are  inserted  in  the  given 
formula,  we  have  : 

-D      96  +  2       98  .  , 

P        ~TZ~  =  IT  =     diametral  pitch. 

When  the  outside  diameter  and  the  number  of 
teeth  are  known,  the  pitch  diameter  may  be  found 
by  multiplying  the  outside  diameter  by  the  number 
of  teeth,  and  dividing  the  product  by  the  sum  of 
2  added  to  the  number  of  teeth;  or,  as  a  formula: 

ZXX  N 

"WTz 

Find  the  pitch  diameter  for  the  gear  having  96 
teeth  and  an  outside  diameter  of  14  inches. 

14X96        1344  , 

^==96T2~       ~98~    =  13.714  inches. 

When  it  is  required  to  find  the  center  distance  C 
between  two  gears  in  mesh  with  each  other,  we 
must  first  know  the  pitch  diameters  of,  or  the 
number  of  teeth  in,  the  two  gears.  The  center 


CALCULATING  THE  DIMENSIONS  OF  GEARS    227 

distance  equals  one-half  of  the  sum  of  the  pitch 
diameters  of  the  two  gears  : 

„      D  +  d 
2 

in  which  D  and  d  denote  the  pitch  diameters  in 
the  large  and  small  meshing  gears,  respectively. 

The  pitch  diameters  of  two  gears  equal  9.  5  and  7 
inches,  respectively.  Find  the  center  distance 
between  them  when  in  mesh. 

•„      9.5  +  7      16.5  . 

O  =        ~^  ~~^~  =  o.Zo  inches. 

The  center  distance  is  also  equal  to  the  sum  of 
the  numbers  of  teeth  in  the  two  gears  divided  by 
two  times  the  diametral  pitch;  or,  as  a  formula: 


2P 

in  which  N  and  n  denote  the  numbers  of  teeth  in 
the  meshing  gears. 

As  an  example,  assume  that  the  number  of  teeth 
in  each  two  gears  equals  95  and  75.  The  diametral 
pitch  is  10.  What  is  the  center  distance? 

n      95  +  75        170 
:   2~~X~10   =    "20" 

We  will  now  find  the  dimensions  of  the  tooth 
parts.  The  addendum  (see  Fig.  160)  equals  1 
divided  by  the  diametral  pitch.  Expressed  as  a 
formula: 


in  which  A  =  addendum. 


228         SELF-TAUGHT  MECHANICAL  DRAWING 

What  is  the  addendum  or  height  above  the  pitch 
line  of  a  5  diametral  pitch  gear  tooth? 

A  =4-=  0.2  inch. 
o 

The  dedendum    (see  Fig.  160)    equals  the  ad- 
dendum. 

The  clearance,   c,   equals  0.157  divided  by   the 
diametral  pitch,  or: 

c_0157 
P. 

What  is  the  clearance  at  the  bottom  of  the  gear 
tooth  (see  Fig.  160)  of  a  4  diametral  pitch  gear? 


c  =  ^p    -  0.039  inch. 

The  full  depth  of  the  tooth  equals  the  sum  of  the 
addendum,  dedendum,  and  clearance,  or 

JL       JL   ,   0.157       2J.57 
P      '  P          P  P 

in  which  d '  =  full  depth  of  gear  tooth. 

What  is  the  full  depth  of  a  4  diametral  pitch 
tooth? 


d,=  =.a539inche 

The  thickness  of  a  cut  gear  tooth  at  the  pitch 
line  equals  1.5708  divided  by  the  diametral  pitch; 
or,  as  a  formula : 

T      1.5708 
~1T 

in  which  T  =  thickness  of  tooth  at  pitch  line. 


CALCULATING  THE  DIMENSIONS  OF  GEARS    229 

What  is  the  thickness  at  the  pitch  line  of  a  4 
diametral  pitch  gear  tooth? 


T  ==  ~       ==  0.3927  inch. 

As  a  general  example,  let  it  be  required  to  de- 
termine the  various  dimensions  for  a  pair  of  gears, 
the  one  having  36  and  the  other  27  teeth.  The 
gears  are  of  8  diametral  pitch. 

By  using  the  formulas  given,  we  have  : 

For  the  larger  gear  : 

Pitch  diameter  =  -5   =  -5-  =  4.5  inches. 

r  o 

Outside  diameter  =  ~~w^  -  86  J"  2=  4.75  inches. 

Jr  o 

For  the  smaller  gear: 

n        27 
Pitch  diameter  =  —  =  -—  =  3.375  inches. 

Jr  o 

Outside  diameter  =  —  5~~  =  —  o  —  =  3.625  inches. 
Jr  o 

For  both  gears  : 

Addendum  =  ~  =  -  -  =  0.125  inch. 

f  O 

Dedendum  =    ^  =  -5-  =  0.125  inch. 
Jr         o 

~  0.157      0.157      ftmofi-     v. 

Clearance  =  —  =r~  =  —  ^~~  =  0.0196  inch. 


Full  depth  of  tooth  =  —      =  -~     -  0.  2696  inch. 

X  O 


36  +  27      63      015  . 
Center  distance  =  —-    -   2x  g  ==  16  =  3||  inch. 


230        SELF-TAUGHT  MECHANICAL  DRAWING 

This  concludes  the  required  calculations  neces- 
sary for  a  pair  of  spur  gears. 

Bevel  Gears. — Bevel  gears  are  used  for  trans- 
mitting motion  between  shafts  whose  shafts  are 
not  parallel,  but  whose  center  lines  form  an  angle 
with  each  other.  In  most  cases  this  angle  is  a 


--o— 
FIG.  173. — Diagram  for  Calculation  of  Bevel  Gearing. 

right,  or  90-degree,  angle.  The  formulas  for  the 
dimensions  of  bevel  gears  are  not  as  simple  as 
those  for  spur  gears,  and  an  understanding  of  the 
trigonometrical  functions,  explained  in  Chapter 
VII,  is  necessary,  as  well  as  the  use  of  trigonomet- 
rical tables.  As  bevel  gears  with  a  90-degree  angle 
between  their  center  lines  are  the  most  common, 


CALCULATING  THE  DIMENSIONS  OF  GEARS    231 

formulas  will  be  given  for  this  case  only,  in  the 
following. 

In  Fig.  173  a  pair  of  bevel  gears  are  shown,  the 
dimensions  of  which  are  to  be  determined.  The 
letters  in  the  formulas  below  denote  the  following 
quantities : 

P  =  diametral  pitch, 

Di  =  pitch  diameter  of  large  gear, 

D2  =  pitch  diameter  of  small  gear, 

01  =  outside  diameter  of  large  gear, 

02  =  outside  diameter  of  small  gear, 
NI  =  number  of  teeth  in  large  gear, 
AT2  =  number  of  teeth  in  small  gear, 

NI  =  number  of  teeth  for  which  to  select  cut- 
ter for  large  gear, 

N2'  =  number  of  teeth  for  which  to  select  cut- 
ter for  small  gear, 
flu  &i,  Ci,  02,  &2,  c2,  d  and  e  =  angles  as  shown  in 

Fig.  173. 
A  =  addendum, 

A  +  C  =  dedendum  =  addendum  plus  clearance. 
If  the  pitch  diameter  and  diametral  pitch  are 
known,    the   number   of   teeth   equals    the  pitch 
diameter  multiplied  by  the  diametral  pitch,  or: 
N,  =  D,  X  P 
N2  =  D2  X  P 

If  the  number  of  teeth  and  the  diametral  pitch 
are  known,  the  pitch  diameter  equals  the  number 
of  teeth  divided  by  the  diametral  pitch,  or: 


232        SELF-TAUGHT  MECHANICAL  DRAWING 

Angles  at  and  a2  can  be  determined  if  either  the 
numbers  of  teeth  or  the  pitch  diameters  of  both 
gears  are  known.  The  tangent  for  these  angles, 
the  pitch  cone  angles,  equals  the  number  of  teeth 
in  one  gear  divided  by  the  number  of  teeth  in  the 
other,  or  the  pitch  diameter  in  one  gear  divided  by 
the  pitch  diameter  in  the  other,  according  to  the 
following  formulas  : 


2          , 
tan  a2=  ^  =  ^ 

Angle  a2  also  equals  90°  -  alt 

The  outside  diameter  equals  the  pitch  diameter 
plus  the  quotient  of  2  times  the  cosine  of  at  or  a2, 
respectively,  divided  by  the  diametral  pitch,  or: 

0,= 


Angles  d  and  e  are  determined  by  -the  formulas: 

,       2  sin  a,!       2  sin  a2 
tancZ=     ^~          -~ 

2.314  sin  a^       2.314  sin  a2 
tan  e  •         -^r-  ^ 

Angles  6t,  Ci,  62  and  c2  are  determined  by  the 
formulas  : 

&!    =    O-!    +   d 

Ct  =  di  —  e 
&2  =  a2  +  d 
c  =  a  -  e 


CALCULATING  THE  DIMENSIONS  OF  GEARS    233 

The  number  of  teeth  for  which  the  cutter  for 
cutting  the  teeth  should  be  selected  is  found  as 
follows  : 


COS 


cosa2 

Finally  the  addendum,  dedendum  and  clearance 
are  found  as  in  spur  gears. 

As  a  practical  example,  assume  now  that  two 
bevel  gears  are  required,  8  diametral  pitch,  with 
24  and  36  teeth,  respectively.  Find  the  various 
dimensions. 

n        Ni       36 

D1  =    -p  --    ~^   =4.5  inches. 

N2       24 
D2=  -p-  =    -g-   =  3  inches. 

tan  ttl  ==  ^  ==  H  =  1.5;  a,  =  56°  20'. 

tan  a2=  ^  =  ^  =  0.667;  a2=  33°  40'. 
JMi       ob 

Qi=Di+  2^0,  =45+  2X|554  =  46W  incheg_ 
02=  Z)2+  ^f^  =  3  +  ^^2  _  3  m  incheg_ 

-L  O 


tan  d  = 


234 


SELF-TAUGHT   MECHANICAL  DRAWING 


b,  =  a,  +  d  =  56°  20'  4-  2°  40'  =  59°  0'. 
Cl  =  a,-  e  =  56020'-3°0'  =  53°20'. 
62  =  a2  +  d  =  33°  40'  +  2°  40'  =  36°  20'. 
c2  =  a2  -  e  =  33  °  40 '  -  3  °  0  '  =  30  °  40 '. 

ZV'=   N*  =  -36 

1    COS  ttj 


AT2'  = 


0.554 
N2     24 
cos  a2   0.832 


=  65  approximately. 
=  29  approximately. 


A  -  -4-  -  4-  -  0.125  inch. 

Jr  o 


=  00196inch 

o 

Whole  depth  of  tooth  =  y  4-  y  + 
0.2696  inch. 


Worm    Gearing.—  Worms   and  worm   gears  are 
used  for  transmitting  power  in  cases  where  great 


FIG.  174. -Worm. 

reduction  in  velocity  and  smoothness  of  action  are 
desired.     They  are  also  used  when  a  self -locking 


CALCULATING  THE  DIMENSIONS  OF  GEARS    235 

power  transmission  is  desirable,  that  is,  when 
it  is  required  that  the  mechanism  itself,  due  to 
the  friction  between  the  worm  and  worm-wheel, 
should  support  the  load  without  slipping  if  the 
driving  power  be  rendered  inoperative. 

In  Figs.  174  is  shown  a  worm  and  in  Fig.  175  a 
worm-wheel;  the  dimensions  to  be  found  are,  in 
most  cases,  given  in  these  illustrations.  The  fol- 
lowing notation  has  been  used  in  the  formulas 
given  below  for  worm  and  worm-wheels : 

P  =  circular  pitch  of  worm-wheel  =  pitch  of  the 

worm  thread, 

N  =  number  of  teeth  in  worm-wheel, 
Z>!  =  pitch  diameter  of  worm-wheel, 
DT  =  throat  diameter  of  worm-wheel, 

01  =  outside  diameter  of  worm-wheel  (to  sharp 

corners) , 

R  =  radius  of  worm-wheel  throat, 

C  =  center  distance  between  worm  and  worm- 
wheel  axes, 

D2  =  pitch  diameter  of  worm, 

02  =  outside  diameter  of  worm, 
DR  =  root  diameter  of  worm, 

A   =  addendum,  or  height  of  worm  tooth  above 

pitch  line, 

d    =  depth  of  worm  tooth, 
a    =  face  angle  of  worm-wheel. 

If  the  pitch  of  the  worm  and  the  number  of 
teeth  in  the  worm-wheel  are  known,  the  pitch 
diameter  of  the  worm-wheel  may  be  found  by 
multiplying  the  pitch  of  the  worm  by  the  number 


236         SELF-TAUGHT  MECHANICAL  DRAWING 


of  teeth,  and  dividing  the  result  by  3.1416,  or,  as 
a  formula  : 

D- 


3.1416 

The  outside  diameter  of  the  worm,  02,  is  usually 
assumed.    To  find  the  pitch  diameter  of  the  worm, 

the  addendum  must  first 
be  found.  The  addendum 
equals  the  pitch  of  the 
worm  thread  multiplied 
by  0.3183,  or: 

A  =  P  X  0.3183. 

Now  the  pitch  diameter 
of  the  worm  equals  the 
outside  diameter  minus  2 
times  the  addendum,  or: 
£>2  =  02  -  2A. 

The  root  diameter  of 
the  worm  can  be  found 
first  after  the  full  depth 
of  the  worm-wheel  thread 
has  been  found.  The  full 
depth  of  the  worm-wheel 
thread  equals  the  pitch 
multiplied  by  0.6866,  or: 
d  =  PX  0.6866. 

Now  the  root  diameter 
of  the  worm  thread  equals 
the  outside  diameter  of 
the  worm  minus  2  times  the  depth  of  the  thread,  or: 
DR  =  02  -  2d. 


FIG.  175.— Worm-wheel. 


CALCULATING  THE  DIMENSIONS  OF  GEARS    237 

The  throat  diameter  of  the  worm-wheel  is  found 
by  adding  2  times  the  addendum  of  the  worm 
thread  to  the  pitch  diameter  of  the  worm-wheel,  or: 

DT  =  A  +  2A. 

The  radius  of  the  worm-wheel  throat  is  found 
by  subtracting  2  times  the  addendum  from  the 
outside  diameter  of  the  worm  divided  by  2,  or: 

R  =    Y2-  2A. 

The  outside  diameter  of  the  worm-wheel  (to 
sharp  corners)  is  found  by  the  formula  below  : 

G!  =  DT  +  2  (R  -  tfcos- 

The  angle  a  is  usually  75  degrees. 

Finally,  the  center  distance  between  the  center 
of  the  worm  and  the  center  of  the  worm-wheel 
equals  the  sum  of  the  pitch  diameter  of  the  worm 
plus  the  pitch  diameter  of  the  worm  gear,  and  this 
sum  divided  by  2,  or: 


Find,  for  an  example,  the  required  dimensions 
for  a  worm  and  worm-wheel,  in  which  the  worm- 
wheel  has  36  teeth,  the  pitch  of  the  worm  thread 
is  J  inch,  and  the  outside  diameter  of  the  worm  is 
3  inches.  We  have  given  P  =  J;  N  =  36;  02  =  3. 

X  36 


3.1416        3.1416  = 

A    =  P  X  0.3183  =  i  X  0.3183  =  0.15915  inch. 
D2  =  02  -  2A  =  3  -  0.3183  =  2.6817  inches. 


238         SELF-TAUGHT  MECHANICAL  DRAWING 

d    =  P  X  0.6866  =  i  X  0.6866  =  0.3433  inch. 

DR  =  02  ~  2d  =  3  -  0.6866  =  2.3134  inches. 

DT  =  D,  +  2A  =  5.730  +  0.3183  =  6.0483  inches. 

R=^-2A=~-  0.3183  =  1.1817  inch. 

0,=  DT  +  2  (R  -  R  cos  y)  ==  6.0483  +  2  X 

(1.1817  -  1.1817  X  cos  37°  30X)  =  6.5375  inches. 

r      A+A       5.730  +  2.6817 

C  =       ~—  — —      -  =  4.2058  inches. 


CHAPTER  XIV 


CONE  PULLEYS 

WHEN  it  is  desired  to  have  a  variable  speed  ratio 
between  two  shafts  which  are  belted  together,  the 
method  of  having  reversed  conical  cylinders  or 
drums  mounted  on  the  shafts,  as  shown  in  Fig.  176 
and  177,  is  sometimes  used.  These  permit  any 


FIG.  176. -Simplest 
Form  of  ' '  Cone- 
Pulley." 


FIG.  177.— An  Im- 
proved Form  of 
"Cone-Pulley." 


FIG.  178.— The  Mod- 
ern Type  of  Stepped 
Cone  Pulley. 


desired  change  of  speed,  but  they  have  disadvan- 
tages which  on  most  work  offset  this  advantage. 
It  would  be  necessary,  in  the  first  place,  to  use  a 
narrow  belt  to  avoid  undue  stretching  at  the  edges. 
Then,  as  the  tendency  of  a  belt  is  to  mount  to  the 
largest  part  of  a  pulley,  this  tendency,  acting  in 

239 


240         SELF-TAUGHT  MECHANICAL  DRAWING 

the  same  way  on  the  cones,  would  produce  undue 
tension  on  the  belt.  If  a  crossed  belt  is  used  on 
such  cones  their  faces  would  be  made  straight,  as 
the  belt  would  be  equally  tight  in  any  position. 
This  may  be  seen  by  an  inspection  of  Fig.  179, 
where  circles  A  and  B  represent  sections  of  such 
cones  on  one  line,  and  circles  C  and  D  represent 
sections  on  another  line.  If  the  cones  have  the 


FIG.  179. — Diagram  Showing  relative  Influence  of  Open  and 
Crossed  Belt  on  Pulley  Sizes. 

same  taper  it  is  evident  that  the  circle  D  will  be 
as  much  larger  than  B  as  C  is  smaller  than  A,  the 
gain  in  one  diameter  being  offset  by  the  loss  in  the 
other.  Then,  as  the  circumferences  of  circles  vary 
directly  as  their  diameters  (the  circumference  of 
a  circle  having  twice  the  diameter  of  another,  for 
instance,  will  be  twice  as  long  as  the  circumfer- 
ence of  the  other) ,  whatever  is  gained  on  one  cir- 
cumference_will  be  lost  on  the  other.  For  a  crossed 
belt  then,  it  is  only  necessary  that  the  cones  have 
the  same  taper. 

When,  however,  an  open  belt  is  used,  it  becomes 
necessary  to  have  the  cones  slightly  bulging  in  the 


CONE  PULLEYS  241 

middle  as  shown  in  Fig.  177.  By  again  inspecting 
Fig.  179  it  will  be  seen  that  it  is  only  when  the 
belt  is  crossed  that  one  cone  gains  as  fast  in  size 
as  the  other  loses,  because  it  is  only  when  the  belt 
is  crossed  that  the  arc  of  contact  of  the  belt  on  the 
pulleys  is  the  same  on  all  steps  of  the  cone. 

In  practice  these  cones  are  usually  replaced  by 
stepped  or  cone  pulleys  as  shown  in  Fig.  178,  so  as 
to  avoid  the  troubles  with  the  belt  previously 
mentioned. 

Applying  the  principles  mentioned  to  cone  pul- 
leys, we  see  that  when  a  crossed  belt  is  used,  all 
that  is  necessary  is  that  the  sum  of  the  diameters 
of  any  pair  of  steps  shall  be  equal  to  the  sum  of 
the  diameters  of  any  other  pair  of  steps.  For 
instance,  the  sum  of  the  diameters  of  steps  1  and 
r  must  be  equal  to  the  sum  of  the  diameters  of 
steps  2  and  2'.  When,  however,  an  open  belt  is 
used,  as  is  usually  the  case,  the  sum  of  the  diam- 
eters of  the  steps  at  or  near  the  middle  of  the  cone 
will  have  to  be  somewhat  greater  than  the  sum  of 
the  diameters  of  those  at  or  near  the  ends. 

What  is  generally  considered  to  be  the  best 
method  of  determining  the  size  of  the  various 
steps  of  cone  pulleys  is  that  given  by  Mr.  C.  A. 
Smith  in  the  '  'Transactions  of  the  American  Society 
of  Mechanical  Engineers, ' '  Vol.  X,  page  269.  Make 
the  distance  C,  Fig.  180,  equal  to  the  distance 
between  the  centers  of  the  shafts,  and  draw  the 
circles  A  and  B  equal  to  the  diameters  of  a  known 
pair  of  steps  on  the  cones.  At  a  point  midway 
between  the  shaft  centers  erect  the  perpendicular 
ab.  Then,  with  a  center  on  ab  at  a  distance  from 


242         SELF-TAUGHT  MECHANICAL  DRAWING 

a  equal  to  the  length  of  C  multiplied  by  0.314,  draw 
the  arc  c  tangent  to  the  belt  line  of  the  given  pair 
of  steps.  The  belt  line  of  any  other  pair  of  steps 
will  then  be  tangent  to  this  arc. 

If  the  angle  which  the  belt  makes  with  the  line 
of  centers,  de,  exceeds  18  degrees,  however,  a 
slight  modification  of  the  above  is  made  as  follows : 
Draw  a  line  tangent  to  the  arc  at  c  at  an  angle 
of  18  degrees  with  de',  and  with  a  center  on  a&,  at 


FIG.  180.— Method  of  Laying  out  Cone  Pulleys. 

a  distance  from  a  equal  to  the  length  C  multiplied 
by  0.298  draw  an  arc  tangent  to  this  18-degree 
line. 

All  belt  lines  which  make  an  angle  with  de 
greater  than  18  degrees  are  made  tangent  to  this 
new  arc. 

The  sizes  of  the  steps  so  obtained  maybe  verified 
by  measuring  the  belt  lengths  of  each  pair.  For 
this  purpose  a  fine  wire  may  be  used,  the  wire 
being  held  in  place  by  pins  placed  at  close  intervals 
on  the  outer  half  circumference  of  each  pulley  of 
the  pair. 


CHAPTER  XV 

BOLTS,  STUDS  AND  SCREWS 

SCREWS  for  clamping  work  together  are  of  three 
classes:  through  bolts,  Fig.  181;  studs,  Fig.  182; 
cap  screws,  Fig.  183.  In  Fig.  181  the  bolt  is  put 
entirely  through  both  of  the  two  pieces  to  be 


FIG.  181.— Through  Bolt  for 
Holding  two  Pieces  to- 
gether. 


FIG.  182.— Stud  used  for 
Clamping  one  Piece  to 
another. 


clamped  together,  and  a  nut  is  put  onto  the 
threaded  end.  This  is  considered  to  be  the  best 
method  on  cast  iron  work,  both  as  regards  efficiency 
and  cheapness,  as  there  is  no  tapping  of  any  holes 

243 


244         SELF-TAUGHT  MECHANICAL  DRAWING 


in  the  cast  iron.  A  tapped  hole  in  cast  iron  is  to 
be  avoided,  if  possible,  as,  on  account  of  the  brittle 
nature  of  the  material,  the  threads  are  liable  to 
crumble  or  wear  away  easily. 

In  many  cases,  however,  it  is  not  practicable  to 
avoid  tapping  holes  in  cast  iron,  or  questions  of 
appearance  may  make  the  broad  flange  which  is 
necessary  when  through  bolts  are  used,  undesirable. 
In  such  cases  studs  should  be  used.  A  stud  consists 
of  a  piece  of  round  stock  threaded  on  both  ends, 

and  having  a  plain  portion 
in  the  middle.  The  studs 
are  screwed  firmly  into  the 
tapped  holes,  which  should 
be  deep  enough  to  prevent 
the  studs  from  bottoming 
in  them,  the  studs  instead 
binding  or  coming  to  a 
bearing  at  the  end  of  the 
threaded  portion.  The 
loose  piece  is  then  put  on 
over  the  studs,  and  is  held 
in  place  by  the  nuts.  By 

using  studs,  any  further  wear  of  the  tapped  hole 
is  avoided,  as,  when  removing  the  loose  part,  the 
nuts  only  are  taken  off,  the  studs  being  left  in 
the  body  piece. 

When  the  material  of  the  parts  which  are  being 
clamped  together  is  of  such  a  nature  that  threads 
formed  in  it  are  not  liable  to  crumble  or  to  rapid 
wear,  then  cap  screws,  Fig.  183,  may  be  used  to 
advantage.  They  give  a  neat  appearance  to  a 
piece  of  work,  and  the  nut  is  entirely  eliminated. 


FIG.  183.— Cap  Screw  used 
for  Clamping  Purposes. 


BOLTS,    STUDS  AND  SCREWS  245 

United  States    Standard    Screw    Thread.— The 

most  commonly  used  of  all  screw  threads  is  the 
United  States  standard  thread.  A  section,  indicat- 
ing the  form  of  this  thread,  is  shown  in  Fig.  184. 
The  thread  is  not  sharp  neither  at  the  top  nor  at 
the  bottom,  but  is  provided  with  a  flat  at  both  of 
these  points,  the  width  of  the  flat  being  one-eighth 
of  the  pitch  of  the  thread.  The  sides  of  the  thread 


x^ 


FIG.  184.— Form  of  the  United  States  Standard  Thread. 

form  an  angle  of  60  degrees  with  each  other.  The 
"pitch"  and  the  "number  of  threads  per  inch" 
should  not  be  confused.  The  pitch  is  the  distance 
from  the  top  of  one  thread  to  the  top  of  the  next. 
If  the  number  of  threads  is  8  per  inch,  then  the 
pitch  would  be  4  inch ;  and  the  flat  on  the  top  of 
a  United  States  standard  thread,  which,  as  men- 
tioned, is  one-eighth  of  the  pitch,  would  be  1-64 
inch.  If  the  number  of  threads  per  inch  is  known, 
the  pitch  may  be  found  by  dividing  1  by  the  num- 
ber of  threads  per  inch,  or 

No.  of  threads  per  inch. 

If,  again,  the  pitch  is  known  and  the  number  of 
threads  per  inch  required,  then 

No.  of  threads  per  inch  =  p.    , 


246         SELF-TAUGHT  MECHANICAL  DRAWING 
U.  S.   STANDARD  SCREW  THREADS. 


BOLTS  AND  THREADS 

HEX.  NUTS  AND 
HEADS. 

SQUARK 
NUT  AND 
HEAD. 

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BOLTS,    STUDS  AND  SCREWS  247 

For  example,  assume  that  the   pitch  is  0.0625 
inch.     Then 

No.  of  threads  per  inch  =  .  =  16. 


The  accompanying  table  of  United  States  stand- 
ard screw  threads  gives  the  standard  number  of 
threads  per  inch,  corresponding  to  given  diame- 
ters, the  diameter  at  the  root  of  the  thread,  the 
width  of  the  flat  at  the  top  and  bottom  of  the 
thread,  the  area  of  the  full  bolt  body,  and  the 
area  at  the  bottom  of  the  thread.  These  dimen- 
sions are,  of  course,  always  the  same  with  all 
manufacturers.  As  regards  the  sizes  for  hexagon 
nuts  and  heads,  and  square  nuts  and  heads  also 
given  in  the  table,  it  may  be  said  that  all  makers 
do  not  conform  strictly  to  the  sizes  as  given.  The 
catalog  of  one  large  bolt  manufacturing  concern, 
which  is  at  hand,  gives  the  width  across  flats  of 
finished  bolt  heads  and  nuts  the  same  as  the  rough 
sizes  given  in  the  table,  which,  it  will  be  seen,  are 
founded  on  the  rule  that  the  width  across  the  flats 
of  the  heads  and  nuts  should  equal  one  and  one- 
half  times  the  diameter  of  the  body  of  the  bolt, 
plus  one-eighth  of  an  inch.  It  will  also  be  noticed 
that  the  thickness  of  the  head  or  nut  is  the  same 
as  the  diameter  of  the  body  of  the  bolt. 

With  cap  screws,  although  the  length  of  the  head 
is  made  the  same  as  for  bolts,  or  equal  to  the  di- 
ameter of  the  bolt  body,  the  diameter  of  the  head, 
and  the  distance  across  flats,  is  made  different  as 
shown  in  table  on  the  following  page  : 


248 


SELF-TAUGHT  MECHANICAL  DRAWING 


CAP   SCREW   SIZES. 
(From  catalog  of  Boston  Bolt  Co. ) 


Size  of  Screw 

I 

T7* 
I 

T56 

i 

TV 

1 
A 

I 

A 

A 
& 

\- 
I 
f 

A 

it 
H 

f 
1 
t 

1 

1 

i 

Width  Across  Flats 
Hex.  Head 

i 
1 

i* 
i* 

it 

H 

Width  Across  Flats 
Square  Head 

Check  or  Lock  Nuts. — When  a  bolt  is  subjected 
to  constant  vibrations  there  is  a  tendency  for  the 
nut  to  work  loose.  To  overcome  this  tendency  it 
is  customary  to  employ  a  second  nut,  called  a  check 
or  lock  nut,  which  is  screwed  down  upon  the  first 
one  as  shown  in  Fig.  185.  When  the  first  nut  is 

screwed  down  to  a  bear- 
ing, the  upper  surfaces  of 
its  thread  are  in  contact 
with  the  under  surfaces 
of  the  bolt  thread.  When 
the  check  nut  is  screwed 
down,  however,  it  forces 
the  first  nut -down  so  that 
the  under  surfaces  of  its 
thread  come  into  contact 
with  the  upper  thread  sur- 
faces of  the  bolt.  This 

means  that  the  check  nut  has  to  bear  the  entire  load. 
When,  therefore,  the  two  nuts  are  of  unequal 
thickness,  as  is  frequently  the  case,  the  thick 
nut  should  be  on  the  outside. 

Bolts  to  Withstand  Shock.— When  a  bolt  which 
is  subjected  to  shocks  fails,  it  breaks,  of  course, 


FIG.  185.— Correct  Arrange- 
ment when  Using  Check 
or  Lock  Nut. 


BOLTS,    STUDS  AND  SCREWS  249 

at  the  part  having  the  least  cross  sectional  area, 
that  is,  at  the  bottom  of  the  thread.  If  now  the 
body  of  the  bolt  be  reduced  so  that  its  cross  section 
is  of  the  same  area  as  the  area  at  the  bottom  of 
the  thread,  a  slight  element  of  elasticity  is  intro- 
duced, and  the  bolt  is  likely  to  yield  somewhat 
instead  of  breaking.  This  is  considered  very  im- 
portant in  some  classes  of  work.  The  reduction  of 
area  may  be  accomplished  by  turning  down  the 
body  of  the  bolt,  or,  according  to  some  authorities, 
the  same  object  is  attained  by  removing  stock  from 
the  inside  by  drilling  into  the  bolt  from  the  head 
end. 

Either  method,  it  is  stated,  gives  the  same  degree 
of  elasticity  to  the  bolt,  but  as  the  drilling  method 
takes  the  stock  from  the  center,  the  bolt  is  left 
stiffer  to  resist  bending  or  twisting  than  when  the 
stock  is  taken  off  the  outside  by  turning. 

Wrench  Action. — When  bolts  or  any  form  of 
screws  are  used  to  hold  machine  parts  together, 
they  must  be  strong  enough  not  only  to  withstand 
the  strain  which  is  put  upon  them  by  the  operation 
of  the  machine,  but  also  to  withstand  the  strain 
which  is  put  upon  them  by  the  wrench  in  setting 
or  screwing  them  up.  In  the  case  of  a  cylinder 
head,  for  instance,  the  strain  upon  the  bolts  due  to 
the  working  of  the  engine  will  be  the  exposed  area 
of  the  head,  multiplied  by  the  pressure  per  square 
inch.  This  divided  by  the  number  of  bolts  used 
will  give  the  proportional  part  of  this  strain  which 
each  bolt  must  sustain.  But  in  order  to  insure  a 
tight  joint,  it  is  necessary  that  the  bolts  be  not 
merely  brought  up  to  a  bearing,  but  that  they  be 


250         SELF-TAUGHT  MECHANICAL  DRAWING 

set  up  hard  enough  so  as  to  press  the  cylinder  and 
cylinder  head  surfaces  firmly  together.  The  force 
which  the  wrench  exerts  in  doing  this  work  will 
be  equal  to  the  circumference  of  the  circle  through 
which  the  hand  moves  in  turning  the  wrench 
through  one  revolution,  multiplied  by  the  force  in 
pounds  exerted  at  the  handle,  and  this  product 
divided  by  the  distance  through  which  the  nut 
advances  in  one  revolution,  that  is,  by  the  lead  of 
the  screw.  This  theoretical  result  is,  of  course, 
modified  by  the  friction  between  the  nut  and  the 
bolt,  and  between  the  nut  and  washer.  In  addition 
to  this  direct  strain,  there  is  also  a  twisting  strain 
in  the  bolt,  caused  by  the  friction  between  the  bolt 
and  nut. 

To  insure  the  bolts  being  sufficiently  strong  to 
resist  these  various  forces,  it  is  customary  to  make 
them  somewhat  more  than  double  the  strength 
that  would  be  necessary  to  enable  them  to  safely 
resist  the  pressure  of  the  steam  or  other  fluid  in 
the  cylinder;  that  is,  they  are  made  about  double 
strength  to  enable  them  to  resist  the  direct  strain 
of  the  wrench  action,  and  then  this  amount  is  in- 
creased about  15  or  20  per  cent,  to  allow  for  the 
twisting  action  of  the  wrench.  Allowing  that  a 
factor  of  safety  of  4  would  be  sufficient  to  allow 
for  the  steam  pressure  only,  a  factor  of  safety  of 
not  less  than  about  9  or  10  would  therefore  be  used 
to  provide  for  the  added  strain  on  the  bolt  due  to 
the  wrench  action.  In  the  case  of  small  bolts, 
where  the  workman  might  set  them  up  much  harder 
than  is  really  necessary,  a  factor  of  safety  of  about 
15  may  be  used. 


BOLTS,   STUDS  AND  SCREWS 


251 


The  distance  apart  which  bolts  can  be  spaced 
without  danger  of  leakage  is  given  by  Prof.  A.  W. 
Smith  as  between  4  or  5  times  the  thickness  of  the 
cylinder  flange  for  pressures  between  100  and  150 
pounds  per  square  inch. 

In  the  case  of  bolts  which  are  not  under  strain 
as  a  result  of  the  wrench  action,  as  in  the  case  of 


FIG.  186.— Example  of  Thread 
not  under  Stress  due  to 
Wrench  Action. 


FIG.  187.— Square  Threaded 
Screw,  such  as  is  Generally 
used  for  Power  Transmis- 


sion. 


the  hook  bolt  shown  in  Fig.  186,  a  factor  of  safety 
as  low  as  4  might  be  properly  used,  if  the  load  is 
steady. 

Assuming  that  the  material  of  which  the  bolts 
are  made  has  an  ultimate  strength  of  40,000  to 
60,000  pounds  per  square  inch,  the  factors  of 
safety  previously  indicated  would  give  allowable 
working  stresses  of  from  4000  to  15,000  pounds  per 
square  inch. 


252        SELF-TAUGHT  MECHANICAL  DRAWING 

Screws  for  Power  Transmission. — In  Fig.  187  is 
shown  a  square  threaded  screw  such  as  is  generally 
used  for  power  transmission.  In  such  a  screw  the 
depth  of  the  thread  is  made  one-half  of  the  pitch. 
The  size  of  the  body  of  the  screw,  assuming  that 
the  work  which  the  screw  is  doing  brings  a  ten- 
sional  stress  on  the  screw,  will  be  determined  by 
the  tensile  strength  of  the  material  of  which  it  is 
made  and  the  factor  of  safety  which  is  used.  As 
a  screw  which  is  used  for  power  transmission  is 
subjected  to  constant  wear  when  in  use,  the  ques- 
tion of  the  proper  amount  of  bearing  surface  in  the 
threads  of  the  nut  is  of  first  importance,  in  order 
that  it  may  not  wear  out  too  rapidly.  The  area  of 
the  thread  surface  in  the  nut  on  which  the  pressure 
bears  will  be  equal  to  the  difference  in  area  of  a 
circle  of  a  diameter  equal  to  the  outside  diameter 
of  the  screw,  and  one  of  a  diameter  equal  to  the 
diameter  at  the  root  of  the  thread  of  the  screw, 
multiplied  by  the  number  of  threads ;  or,  letting  D 
represent  the  outside  diameter  of  the  screw,  and  d 
represent  the  diameter  of  the  body, the  area  will  be: 

(D2  -  d2)  X  0.7854  X  No.  of  threads  in  the  nut. 

The  allowable  pressure  per  square  inch  of  working 
surface  will  vary  with  the  nature  of  the  service 
required,  whether  fast  or  slow,  and  also  with  the 
lubrication,  and  with  the  material  used.  Where 
the  speed  is  slow,  say  not  over  50  feet  per  minute, 
and  the  service  is  infrequent,  as  in  lifting  screws, 
a  pressure  of  2500  pounds  for  iron  or  3000  pounds 
for  steel  is  allowable,  while  for  more  constant 
service  some  authorities  limit  the  pressure  to 
about  1000  pounds  per  square  inch  even  when  the 


BOLTS,   STUDS  AND  SCREWS  253 

lubrication  is  good.  For  high  speeds  a  pressure  of 
about  200  or  250  pounds  is  considered  to  be  as 
much  as  should  be  allowed. 

For  a  screw  which,  fitting  loosely  in  a  well  lubri- 
cated nut,  is  to  sustain  a  load  without  danger  of 
running  down  of  itself,  the  pitch  of  the  screw 
should  not,  according  to  Professor  Smith,  be  greater 
than  about  one-tenth  of  its  circumference. 

Efficiency  of  Screws. — A  square-threaded  screw 
has  a  greater  efficiency  than  a  V- threaded  one,  as 
the  sloping  sides  of  the  V-thread  cause  an  increase 
of  friction.  Square  threads  are  therefore  preferable 
for  power  transmission.  Experiments  show  that 
in  the  case  of  bolts  used  for  fastenings,  the  friction 
of  the  nut  on  the  bolt  and  washer  may  absorb  90 
per  cent,  of  the  power  applied  to  the  wrench, 
leaving  only  10  per  cent,  for  producing  direct  com- 
pression. For  square-threaded  screws  an  efficiency 
of  about  50  per  cent,  is  considered  fair,  if  the 
screws  are  well  lubricated. 

Acme  Standard  Thread. — While  the  square  thread 
gives  the  greatest  efficiency  in  a  screw  it  is  not  as 
strong  as  one  having  sloping  sides.  Fig.  188  shows 
a  section  of  a  screw  thread  called  the  Acme  or  29- 
degree  thread,  which  is  often  used  for  replacing 
the  square  thread  for  many  purposes,  such  as  in 
screws  for  screw  presses,  valve  stems,  and  the 
like.  The  use  of  such  a  screw  permits  the  employ- 
ment of  a  split  nut,  when  such  construction  is 
desirable,  which  would  not  be  practicable  with  a 
perfectly  square  thread,  and  for  this  reason,  as 
well  as  for  the  reason  that  it  can  be  cut  with 
greater  ease  than  the  square  thread,  it  has  of  late 


254         SELF-TAUGHT  MECHANICAL  DRAWING 

become  widely  used.  In  the  Acme  standard  thread 
system  the  threads  on  the  screw  and  in  the  nut  are 
not  exactly  alike.  A  clearance  of  0.010  inch  is 
provided  at  the  top  and  at  the  bottom  of  the  thread, 
so  that  if  the  screw  is  1  inch  in  diameter,  for 
example,  then  the  largest  diameter  of  the  thread 
in  the  nut  would  be  1.020  inch.  If  the  root  diam- 
eter of  the  same  screw  were  0.900  inch,  then 
the  smallest  diameter  of  the  thread  in  the  nut 
would  be  0.920  inch.  The  sides  of  the  threads, 
however,  fit  perfectly. 

The  depth  of  an  Acme  thread  equals  one-half  the 
pitch  of  the  thread  plus  0.010  inch.     The  width 


FIG.  188. -Shape  of  Acme  Screw  Thread. 

of  the  flat  at  the  top  of  the  screw  thread  equals 
0.3707  times  the  pitch;  and  the  width  of  the  flat 
at  the  bottom  of  the  thread  equals  0.3707  times  the 
pitch  minus  0.0052  inch. 

Miscellaneous  Screw  Thread  Systems. — Besides 
the  screw  thread  systems  already  mentioned,  a 
great  many  other  systems  are  in  more  or  less 
common  use.  Leading  among  these  is  the  sharp 
V-thread,  which,  previous  to  the  introduction  of  the 
United  States  standard  thread,  was  the  most  com- 
monly used  thread  in  this  country.  This  thread 
is,  theoretically  at  least,  sharp  at  both  the  top  and 
the  bottom  of  the  thread,  the  angle  between  the 


BOLTS,    STUDS  AND  SCREWS  255 

sides  of  the  thread  being  the  same  as  in  the  United 
States  standard  system,  or  60  degrees.  In  ordinary 
practice,  however,  a  small  flat  is  provided  on  the 
top  of  the  thread,  because  it  would  be  almost  impos- 
sible to  commercially  produce  the  thread  otherwise ; 
and  even  if  the  thread  could  be  produced,  the  sharp 
edge  at  the  top  would  rapidly  wear  away.  The 
sharp  V-thread  is  being  more  and  more  forced 
out  of  use  by  the  United  States  standard  thread, 
although  it  must  be  admitted  that  it  will  probably 
long  hold  its  own  in  steam  fitting  work,  because  of 
being  especially  adapted  for  making  steam-tight 
joints.  It  answers  this  purpose  probably  better 
than  any  of  the  other  common  forms  of  threads. 

The  Whitworth  standard  thread  is  not  used  to  a 
very  great  extent  in  the  United  States,  but  it  is  the 
recognized  standard  thread  in  Great  Britain.  In 
this  form  of  thread  the  sides  of  the  thread  form 
an  angle  of  55  degrees  with  each  other,  and  the 
tops  and  bottoms  of  the  threads  are  rounded  to  a 
radius  equal  to  0.137  times  the  pitch.  This  round- 
ing of  the  thread  at  the  top  provides  for  a  thread 
which  does  not  wear  rapidly,  and  screws  and  nuts 
made  according  to  this  thread  system  will  work 
well  together  in  continuous  heavy  service  for  a 
longer  period  than  would  screws  and  nuts  with  any 
of  the  other  standard  thread  forms.  The  fact  that 
the  threads  are  rounded  in  the  bottom  is  advan- 
tageous on  account  of  the  elimination  of  sharp 
corners  from  which  fractures  may  start.  The  main 
disadvantage  of  the  thread,  and  the  reason  why 
the  United  States  standard  thread  was  adopted  in 
this  country  in  preference  to  the  Whitworth  stand- 


256        SELF-TAUGHT  MECHANICAL  DRAWING 

ard,  which  is  the  older  of  the  two,  is  to  be  found 
in  the  fact  that  it  is  more  difficult  to  produce  than 
a  60-degree  thread  with  flat  top  and  bottom.  The 
Whitworth  form  of  thread  is  used  in  this  country 
mostly  on  special  work  and  on  stay-bolts  for  loco- 
motive boilers.  • 

A  thread  perhaps  more  commonly  used  than  any 
of  the  others,  with  the  exception  of  the  United 
States  standard  thread,  is  the  Briggs  standard 
pipe  thread,  which  is  used,  as  the  name  indicates, 
for  pipe  fittings.  This  thread  is  similar  to  the 
sharp  V-thread,  having  an  angle  of  60  degrees 
between  the  sides,  and  nearly  sharp  top  and  bottom ; 
instead  of  being  exactly  sharp  at  the  top  and  bot- 
tom, however,  it  is  slightly  rounded  off  at  these 
points.  The  difficulty  of  producing  these  slightly 
rounded  surfaces  has  brought  about  a  modification, 
at  least  in  the  United  States,  so  that  a  small  flat  is 
made  at  the  top,  and  the  thread  made  to  a  sharp 
point  at  the  bottom.  It  appears  that  a  thread  cut 
with  these  modifications  serves  its  purpose  equally 
as  well  as  a  thread  cut  according  to  the  original 
thread  form. 

Besides  these  systems,  there  are  the  metric  screw 
thread  systems.  These  use  the  same  form  of  thread 
as  the  United  States  standard  system,  but  the 
thread  diameters  and  the  corresponding  pitches 
are,  of  course,  made  according  to  the  metric  system 
of  measurement. 

Other  Commercial  Forms  of  Screws. — Set-screws, 
shown  in  Fig.  189,  are  usually  made  with  square 
heads,  and  have  either  round  or  cup-shaped  points, 
and  are  generally  case  hardened.  They  are  used 


BOLTS,    STUDS  AND  SCREWS 


257 


for  such  work  as  fastening  pulleys  onto  shafts, 
etc.  Some  set-screws  are  made  headless,  and  are 
slotted  for  use  with  a  screw-driver  in  places  where 
it  is  undesirable  that  the  _ 

screw   projects    beyond    the 
work. 

The  term  machine  screws 
covers  a  number  of  styles  of 
small  screws  made   for  use 
with    a    screw-driver.     Fig. 
190  shows  the  principal  styles. 
Machine  screw  sizes  are  usu- 
ally designated  by  numbers, 
the  size  and  the  number  of 
threads  per  inch  being  usually  given  together,  with 
a  "dash"  between ;  thus  a  10 — 24  screw  would  be  a 
number  10  screw  with  24  threads  per  inch.    There 
are    two   standard  systems    for  machine    screw 


FlG.  189.— Forms  of 
Set-screws. 


FILLISTER 
HEAD 


FIG.  190.— Forms  of  Machine  Screws. 

threads,  the  old,  which  until  recently  was  the  only 
system,  and  the  new,  which  was  approved  in  1908 
by  the  American  Society  of  Mechanical  Engineers. 
The  standard  thread  form  of  the  old  sy"stem  was 
the  sharp  V-thread,  with  a  liberal  but  arbitrarily 


258 


SELF-TAUGHT  MECHANICAL  DRAWING 


selected  flat  on  the  top.  The  basic  thread  form 
of  the  new  system  is  that  of  the  United  States 
standard  thread. 

The  accompanying  tables  give  the  numbers  and 
corresponding  diameters  and  number  of  threads 
per  inch  of  the  old  as  well  as  the  new  system  for 
machine  screw  threads. 

MACHINE  SCREW  THREADS,  OLD  SYSTEM. 


Number. 

Diameter. 

Threads 
per  inch. 

Number. 

Diameter. 

Threads 
per  inch. 

1 

0.071 

64 

12 

0.221 

24 

1* 

0.081 

56 

13 

0.234 

22 

2 

0.089 

56 

14 

0.246 

20 

3 

0.101 

48 

15 

0.261 

20 

4 

0.113 

36 

16 

0.272 

18 

5 

0.125 

36 

18 

0.298 

18 

6 

0.141 

32 

20 

0.325 

16 

7 

0.154 

32 

22 

0.350 

16 

8 

0.166 

32 

24 

0.378 

16 

9 

0.180 

30 

26 

0.404 

16 

10 

0.194 

24 

28 

0.430 

14 

11 

0.206 

24 

30 

0.456 

14 

MACHINE  SCREW  THREADS,  NEW  SYSTEM. 


Number. 

Diameter. 

Threads 
per  inch. 

I  Number. 

Diameter. 

Threads 
per  inch. 

0 

0.060 

80 

12 

0.216 

28 

1 

0.073 

72 

14 

0.242 

24 

2 

0.086 

64 

16 

0.268 

22 

3 

0.099 

56 

18 

0.294 

20 

4 

0.112 

48 

20 

0.320 

20 

5 

0.125 

44 

22 

0.346 

18 

6 

0.138 

40 

24 

0.372 

16 

7 

0.151 

36 

26 

0.398 

16 

8 

0.164 

36 

28 

0.424 

14 

9 

0.177             32 

30 

0.450 

14 

10 

0.190 

30 

! 

CHAPTER  XVI 

COUPLINGS  AND  CLUTCHES 

A  COUPLING  is  a  device  for  connecting  together 
the  ends  of  two  shafts  or  axles  for  the  purpose 
of  making  a  longer  shaft,  the  term  being  usually 
limited  to  those  devices  which  are  intended  for 
permanent  fastening.  The  term  clutch  is  used  to 
designate  a  disengaging  coupling. 

The  simplest  form  of  coupling  consists  simply  of 
a  sleeve  or  muff,  made  of  a  length  about  three 
times  the  diameter  of  the  shaft,  bored  out  to  fit 
the  shaft,  and  provided  with  a  keyway  its  entire 
length,  made  to  receive  a  tapering  key.  The  ends 
of  the  shafting  are,  of  course,  also  provided  with 
keyways,  and  are  inserted  into  the  sleeve;  then 
the  key  is  driven  in.  In  some  couplings  the  sleeve 
is  made  tapering  on  the  outside  at  both  ends,  and, 
being  split,  is  clamped  upon  the  shafts  by  means 
of  rings  or  hollow  conical  sleeves  which  are  driven 
onto  the  tapered  ends,  or  drawn  together  by  means 
of  bolts. 

One  of  the  most  common  forms  of  coupling  is  the 
flange  coupling  shown  in  Fig.  191.  In  this  case  a 
flanged  hub  is  keyed  to  each  of  the  shaft  ends,  and 
the  flanges  are  then  held  together  and  prevented 
from  turning  relative  to  each  other  by  bolts,  as 
shown.  In  some  cases  the  bolt  heads  and  nuts  are 

259 


260         SELF-TAUGHT  MECHANICAL  DRAWING 

provided  with  a  guard  by  having  the  rim  on  the 
outer  edge  of  the  flange  made  deep  as  shown  by 
the  dotted  lines  on  one  side.  This  construction 
also  allows  the  coupling  to  be  used  as  a  pulley,  if 
necessary.  In  a  coupling  of  this  kind,  the  chief 
problem  is  to  get  the  bolts  of  such  size  that  their 
combined  strength  to  resist  the  shearing  action 
to  which  they  are  subjected  equals  the  twisting 


FIG.  191.—  Flange  Coupling. 

strength  of  the  shaft.  Letting  d  represent  the 
diameter  of  the  shaft  in  inches,  its  internal  resist- 
ance to  twisting  is  given  by  the  formula 


~  5.1 

in  which  T  equals  the  internal  resistance  to  twist- 
ing, or  the  twisting  moment,  and  S  the  shearing 
strength  per  square  inch  of  area  in  pounds. 

Regarding  the  shearing  strength  of  materials 
Kent  says  :  '  '  The  ultimate  torsional  shearing  re- 
sistance is  about  the  same  as  the  direct  shearing 
resistance,  and  may  be  taken  at  20,000  to  25,000 
pounds  per  square  inch  for  cast  iron,  45,000  pounds 


COUPLINGS  AND  CLUTCHES  261 

for  wrought  iron,  and  50,000  to  150,000  pounds  for 
steel  according  to  its  carbon  and  temper." 

The  torsional  and  direct  shearing  resistance  being 
the  same,  this  quantity  may  be  neglected  if  the 
shaft  and  coupling  bolts  are  of  the  same  material, 
and 

ds 
5.1 

the  internal  resistance  factor  or  torsion  modulus 
of  the  shaft,  should  be  equal  to  the  product  of  the 
radius  of  the  bolt  circle  of  the  coupling,  the  number 
of  bolts  used,  and  the  area  of  each  bolt.  Or,  letting 
a  represent  the  area  of  each  bolt,  R  the  radius  of 
the  bolt  circle  of  the  coupling,  and  n  the  number 
of  bolts  used,  we  would  have : 

a  =  -~-  +  (R  X  ri). 

£).  -L 

Example. — Required  the  size  of  the  bolts  for  a 
flange  coupling  for  a  2-inch  shaft.  The  radius  of 
the  bolt  circle  is  3  inches,  four  bolts  being  used. 

Using  the  notation  in  the  formula  given,  our 
known  values  are: 

d  =  2  inches, 
R  =  3  inches, 
n  =  4  bolts. 

If  we  insert  these  values  in  the  formula  we  have: 

03  o 

a  =  -g-y  -^  (3X4)=  g-^^  12 =0.13  square  inches. 

This  area  corresponds  to  a  diameter  of  about  A 
of  an  inch.  To  allow  for  the  strain  on  the  bolt 
caused  by  the  action  01  the  wrench,  the  next  size 


262        SELF-TAUGHT  MECHANICAL  DRAWING 


larger  bolt,  at  least,  or  a  i  inch  bolt,  will  be  se- 
lected. The  capacity  of  the  bolt  to  resist  shear- 
ing will  be  considerably  increased  by  having  the 
corners  of  the  holes  at  those  faces  of  the  flanges 
which  come  together,  somewhat  rounded.  If  this 
is  not  done,  the  action  of  the  flanges  on  the  bolts 
will  be  like  that  of  a  pair  of  sharp  shears.  Experi- 
ments have  shown  that  with  the  corners  rounded, 
the  capacity  of  the  bolt  to  resist  shearing  may  be 
increased  12  per  cent. 

If  the  shaft  and  bolts  are  of  different  materials 
then  the  modulus 

^ 
5.1 

should  be  multiplied  by  the  shearing  strength  of 
the  shaft  in  pounds  per  square  inch  and  the  product 


t= 


3  d » 

FIG.  192.— Clamp  Coupling. 

RXn  should  be  multiplied  by  the  shearing  strength 
of  the  bolts  per  square  inch,  before  dividing  in 
the  formula  to  get  the  bolt  area. 

In  Fig.  192  is  shown  another  form  of  coupling 
much  used.  It  consists  of  two  parts  bolted  together 
over  the  joint  in  the  shafting,  a  key  and  keyway 
being  provided  to  prevent  the  slipping  of  the  shafts. 


COUPLINGS  AND   CLUTCHES 


263 


By  having  a  thickness  of  heavy  paper  interposed 
between  the  two  parts  of  the  coupling  when  it  is 
bored  out,  it  may  be  made  to  clamp  very  tightly 
onto  the  shafts. 

With  either  form  of  coupling,  the  length  is  made 
such  that  each  shaft  end  is  held  by  the  coupling 
by  a  length  of  about  one  and  one-half  times  its 
diameter,  as  indicated  in  the  engravings. 

Oldham's  Coupling. — Fig.  193  shows  a  form  of 
coupling  which  may  be  used  for  shafts  which  are 


FIG.  193.— Oldham's  Coupling. 

parallel,  but  slightly  out  of  line.  In  this  coupling 
each  shaft  end  has  a  flanged  hub  attached  to  it. 

Across  the  face  of  each  flange  is  planed  a  single 
groove  passing  through  its  center.  Interposed 
between  the  two  flanges  is  a  disk,  shown  at  the 
right,  having  tongues  on  both  faces  at  right  angles 
to  each  other,  to  engage  in  the  grooves  in  the 
flanges. 

Hooke's  Coupling  or  Universal  Joint  is  used  for 
connecting  two  shafts  whose  axes  are  not  in  line 
with  each  other,  but  merely  intersect.  The  shafts 
A  and  B,  and  B  and  C,  in  Fig.  194,  are  thus  con- 
nected by  universal  joints.  If  the  shaft  B  is  made 
telescoping,  as  is  very  often  the  case,  a  solid  part 


264         SELF-TAUGHT  MECHANICAL  DRAWING 

entering  into  and  being  keyed  in  a  sleeve  so  as 
to  prevent  independent  rotation,  but  yet  permit 
a  sliding  action,  then  the  two  shafts  A  and  C  may 
move  independently  of  each  other  within  certain 
limits,  the  distance  between  their  ends  being 
capable  of  variation.  The  arrangement  shown  in 
Fig.  194  is  used  on  various  machine  tools,  notably 
on  milling  machines,  flange  drilling  machines,  etc. 
Many  designs  of  flexible  shafts  are  really  only  a 
combination  of  a  great  number  of  universal  joints. 


FIG.  194.  —  Application  of  Universal  Joints  and  Telescoping 


When  this  coupling  is  employed  for  driving  only 
one  shaft  at  an  angle  with  another,  as  if  shaft 
A  simply  drove  shaft  B  which,  of  course,  is  the 
fundamental  type  of  universal  coupling,  then,  if 
the  driving  shaft  has  a  uniform  motion,  the  driven 
shaft  will  have  a  variable  motion,  and  so  cannot  be 
used  in  such  cases  where  uniformity  of  motion  of 
the  driven  shaft  is  necessary  ;  but  where  there  are 
three  shafts,  as  shown  in  the  illustration,  A  will 
impart  a  uniform  motion  to  C  provided  the  axes  of 
A  and  C  are  parallel  with  each  other,  as  shown  ;  for 
if  A,  having  a  regular  motion,  imparts  an  irregular 


COUPLINGS  AND  CLUTCHES 


265 


motion  to  B,  then  if  B,  with  its  irregular  motion, 
is  made  the  driver,  it  will  impart  a  regular  motion 
to  A,  and  as  C  is  parallel  with  A  it  will  also  impart 
a  regular  motion  to  C. 

This  form  of  coupling  does  not  work  very  well 
if  the  angle  a  is  more  than  45  degrees. 

Clutches  are  of  two  general  classes,  toothed 
clutches  and  friction  clutches.  An  example  of  a 
toothed  clutch  is  shown  in  Fig.  195.  In  this  clutch 
the  part  at  the  left  is  fastened  to  its  shaft;  the 
part  at  the  right  is  free  to  slide  back  and  forth  upon 


FIG.  195.— A  simple  Form  of  Toothed  Clutch. 

its  shaft,  but  is  prevented  from  turning  on  the 
shaft  by  a  key.  The  sliding  motion  for  engaging 
or  disengaging  this  part  of  the  clutch  is  accom- 
plished by  means  of  the  forked  lever  and  jointed 
ring,  shown  at  the  right,  which  latter  engages  in 
the  groove  A.  Such  a  clutch,  while  giving  a  pos- 
itive drive,  cannot,  of  course,  be  thrown  in  or  out 
while  the  driving  shaft  is  running  at  a  high  rate 
of  speed.  By  having  the  back  faces  of  the  teeth 
beveled  off  as  shown  by  the  dotted  lines,  this  diffi- 
culty is  partly  overcome,  although  the  shock  caused 
by  the  sudden  engaging  of  the  teeth  still  renders 


266 


SELF-TAUGHT  MECHANICAL  DRAWING 


the  clutch  unsuitable  for  operating  at  very  high 
speed.  To  facilitate  uncoupling,  the  driving  faces 
may  also  be  given  an  angle  of  about  10  or  12 
degrees. 

Friction  Clutches  are  generally  made  in  one  of 
the  two  styles  shown  in  Figs.  196  and  197.  The 
power  which  a  clutch  of  the  type  shown  in  Fig.  196 
will  transmit,  depends  upon  the  power  which  is  ap- 
plied to  force  the  sliding  part  against  the  fixed  part, 


FIG.  196.— Friction -Disk  Clutc.h. 

and  the  efficiency  of  the  frictional  force  between 
the  rubbing  surfaces.  As  to  the  efficiency  of  the 
clutch,  therefore,  much  depends  upon  the  nature  of 
the  engaging  surfaces,  whether  metal  comes  in 
contact  with  metal,  or  whether  one  of  the  surfaces 
has  a  facing  of  leather  or  wood.  The  efficiency  is, 
of  course,  much  increased  by  either  a  leather  or 
wood  facing.  Professor  Smith  gives  the  efficiency 
of  these  different  surfaces  as  follows:  Cast  iron  on 
cast  iron,  10  to  15  per  cent. ;  cast  iron  on  leather, 


COUPLINGS  AND  CLUTCHES  267 

20  to  30  per  cent. ;   cast  iron  on  wood,  20  to  50  per 
cent. 

The  horse-power  which  such  a  clutch  will  trans- 
mit will  be  found  by  multiplying  the  velocity  of  the 
parts  in  contact,  in  feet  per  minute,  taken  at  their 
mean  diameter  as  indicated  at  D,  by  the  force 
which  is  being  applied  at  this  diameter  in  the 
direction  of  revolution,  and  dividing  this  product 
by  33, 000.  The  force  which  is  acting  at  the  diameter 
D  to  produce  revolving  motion  is  equal  to  the  pres- 
sure which  is  being  applied  to  force  the  two  parts 
of  the  clutch  together,  multiplied  by  the  coefficient 
of  friction  (as  the  frictional  efficiency  between  the 
surfaces  in  contact,  as  given  above,  is  called)  of  the 
materials  which  form  the  driving  surfaces. 

Example. — What  power  will  a  clutch  of  the  type 
shown  in  Fig.  196  transmit  if  running  at  a  speed 
of  250  revolutions  per  minute?  The  diameter  D  is 
18  inches,  and  a  pressure  of  50  pounds  is  exerted 
to  force  the  two  clutch  faces  together.  One  of 
the  clutch  parts  has  a  leather  facing,  and  the 
coefficient  of  friction  is  0.25. 

The  general  formula  for  finding  the  horse-power 
of  a  clutch  of  this  type  is: 

IT  p       £>X3.1416XnXPX/ 

33,000 
in  which  H.P.  =  horse-power  transmitted, 

D  =  mean  diameter  of  friction  sur- 
faces in  feet, 

n  =  revolutions  per  minute, 
P  =  pressu/e  between  clutch  surfaces 

in  pounds, 
/  =  coefficient  of  friction. 


268         SELF-TAUGHT  MECHANICAL  DRAWING 

The  values  to  be  inserted  in  the  formula,  which 
are  given  in  this  problem,  are  as  follows  : 

D  =  ]n  =  1.5  foot, 

n  =  250  revolutions, 
P  =  50  pounds, 
/  =  0.25. 

Inserting  these  values  in  the  formula  we  have  : 

1.5  X  3.1416  X  250  X  50  X  0.25       n  ,_ 
"33,000 

The  formula  given  may  be  transposed  in  various 
ways  according  to  the  requirements  of  the  problem  ; 
if,  for  instance,  it  is  desired  to  know  what  pressure 
must  be  applied  to  transmit  a  given  horse-power, 

then: 

H.P.  X  33,000 

D  (in  feet)"  X  3.1416  X  n  X  /. 

If  the  pressure  is  known,  and  it  is  required  to 
find  what  diameter  the  clutch  must  be  made  to 
transmit  a  given  power,  then  : 


D  (in  feet)  -       - 

~  3.1416  X  n  X  P  X  /. 

If  the  pressure  and  diameter  are  both  known, 
then  the  number  of  revolutions  which  the  clutch 
must  make  per  minute  to  transmit  a  given  horse- 

power will  be  : 

H.P.  X  33,000 


n  = 


D  (in  feet)  X  3.1416  X  P  X  /. 


It  may  be  said  that  the  capacity  of  the  clutch 
to  transmit  power  is  independent  of  the  area  of  the 


COUPLINGS  AND  CLUTCHES 


269 


friction  surfaces;  for,  if  the  friction  surface  is 
increased  the  pressure  which  is  applied  to  force 
the  two  parts  of  the  clutch  together  is  simply  dis- 
tributed over  a  much  greater  area,  giving  a  smaller 
pressure  per  square  inch.  The  durability  would  be 
increased,  but  the  horse-power  capacity  would  re- 
main unchanged. 

The  conical  clutch  shown  in  Fig.  197  may  be 
made  to  run  metal  to  metal,  or  the  hollow  part  may 


FIG.  197.— Friction  Cone  Clutch. 

be  made  larger  to  allow  of  the  insertion  of  wooden 
blocks.  This  would  increase  the  efficiency,  but  at 
the  expense  of  the  durability.  The  principle  of 
this  form  of  clutch  may  be  explained  by  referring 
to  the  diagrammatical  sketch  at  the  right  of  Fig. 
197,  where  the  angle  ACB  represents  the  angle 
which  the  opposite  sides  of  the  clutch  make  with 
each  other,  the  line  DC  representing  the  axis  of 
the  shaft.  If  now  line  bd  of  the  small  triangle 
abd  be  considered  as  representing  the  magnitude 


270         SELF-TAUGHT  MECHANICAL  DRAWING 

of  the  force  acting  in  the  direction  of  the  axis 
of  the  shaft  to  force  the  two  parts  of  the  clutch 
together,  then  if  ab  is  at  right  angles  to  AC,  ab 
will  represent  the  resultant  magnitude  of  the  force 
acting  on  the  face  of  the  clutch  at  right  angles  to 
its  surface,  according  to  the  principles  explained 
in  the  chapter  on  the  elements  of  mechanics.  The 
efficiency  of  the  clutch  will  therefore  be  as  much 
greater  than  that  of  a  flat-faced  clutch  as  ab  is 
greater  than  bd.  The  horse-power  of  such  a 
clutch,  using  the  same  notation  as  before,  would, 
therefore,  be: 

_  D  (in  feet)  X  3.1416  X  n  X  P  Xf  v  ab^ 
33,000  K  bd. 

But  from  the  chapter  on  the  solution  of  triangles 
we  know  that 

— T  =  sine  of  angle  bad. 

Hence 

ab  1 


bd       sin  bad. 

But  angle  bad  equals  angle  x,  the  angle  which 
the  conical  surface  of  the  clutch  makes  with  the 
axis  of  the  shaft. 
Therefore 

ab  =      1 
bd       sin  x 

and  our  original  formula  takes  the  form: 

TJ  P       D  On  feet)  X  3.1416  X  n  X  P  Xf 
'  33,000  X  sin  x. 


COUPLINGS  AND  CLUTCHES  271 

Transposing  this  formula  as  before  for  the  flat- 
faced  clutch,  gives  us: 

H.P.  X  33,000  X  sin  a? 
~  D  (in  feet)  X  3.1416  XnXf. 

nr     -    ,,        H.P.  X  33,000  X  sin  x 
D  (in  feet)  -     ~  " 


n  = -— = ,____^  srn^ 

D  (in  feet)  X  3.1416  X  Pxf. 

The  sine  of  x  may  be  taken  from  the  tables  of 
trigonometric  functions  previously  given  in  the 
chapter  on  the  solution  of  triangles,  or  it  may  be 
found  by  dividing  the  length  bd  (Fig.  197)  by  the 
length  ab. 

The  power  necessary  to  force  the  two  parts  of 
the  clutch  together  may  be  neglected,  as  the  slip- 
ping which  occurs  as  they  are  engaging  allows 
them  to  come  together  with  but  little  pressure 
beyond  what  is  required  for  power  transmission 
purposes.  The  angle  which  the  face  of  the  clutch 
makes  with  the  shaft  (the  angle  x  in  the  diagram 
at  the  right  in  Fig.  197)  should  be  such  that  the 
clutch  does  not  grip  too  quickly  when  thrown  into 
gear,  nor  require  too  much  pull  to  release.  Making 
this  angle  between  7  and  12  degrees  conforms  to 
the  average  given  by  different  authorities. 


CHAPTER  XVII 

SHAFTS,  BELTS  AND  PULLEYS 

Shafts. — The  twisting  strength  of  a  shaft,  as 
stated  in  the  preceding  chapter,  is  given  by  the 
formula 

T  = 


5.1 

in  which  T  =  twisting   moment,    or  force    which 
acting  at  a  distance  of  one  inch 
from  the  center  of  the  shaft  would 
produce  in  it  a  torsional  shearing 
stress  of  S  pounds  per  square  inch, 
d  =  diameter  of  shaft  in  inches, 
S  =  torsional  shearing  stress  in  pounds 
per  square  inch. 

Expressing  this  formula  in  words  we  may  say 
that  the  cube  of  the  diameter  in  inches  multiplied 
by  the  torsional  shearing  stress,  and  this  product 
divided  by  5.1,  gives  the  force  which  acting  at  a 
distance  of  one  inch  from  the  center  of  the  shaft 
would  produce  in  it  the  given  torsional  shearing 
stress. 

The  twisting  moment  T  equals,  therefore,  the 
force  Fl9  acting  at  a  distance  of  one  inch  from 
the  center  of  the  shaft,  times  1 ;  it  also  equals  any 
other  force  F  exerting  a  twisting  action  on  the 

272 


SHAFTS,  BELTS  AND  PULLEYS       273 

shaft  multiplied  by  its  distance  from  the  center  of 
the  shaft.    The  formula  given  can  hence  be  written 


5.1 

vn  which  F  =  any  force  acting  at  a  distance  r  from 
the  center  of  the  shaft. 

Transposing  this  formula  to  obtain  the  distance 
from  the  center  (r)  at  which  a  given  force  would 
have  to  act  to  set  up  a  torsional  shearing  stress  S 
in  the  shaft,  we  would  have: 

r  "=  KlXF. 

The  force  which  would  be  necessary  to  set  up 
a  stress  S  in  the  shaft  when  acting  at  a  given 
distance  would  be : 

=  5.1  Xr. 

The  diameter  of  shaft  to  resist  a  given  force 
acting  at  a  given  distance  would  be: 


d  --  ^F 


r  X5.1 


The  torsional  shearing  strength  of  ordinary 
shafting  is  about  45,000  pounds  to  the  square  inch, 
and  of  steel  shafting  from  about  50,000  to  150,000 
pounds,  according  to  its  quality;  these  figures 
should  be  divided  by  five  or  six  to  give  a  safe 
working  stress. 

The  above  formulas,  however,  are  based  on  the 
assumption  that  the  force  acting  is  of  a  purely 


274         SELF-TAUGHT  MECHANICAL  DRAWING 

twisting  nature,  as  if  a  hand-wheel  were  put  onto 
the  end  of  the  shaft,  and  the  tendency  to  bend  the 
shaft,  caused  by  the  pull  of  one  hand,  were  counter- 
acted by  the  push  of  the  other  hand.  In  the  case 
of  a  shaft  actuated  by  a  rocker  arm,  as  sometimes 
occurs  in  machines,  the  tendency  to  bend  the  shaft 
caused  by  the  push  on  the  arm  could  be  provided 
for  by  using  a  somewhat  higher  factor  of  safety. 
If  the  arm  were  placed  at  some  distance  from  the 
bearing,  however,  the  tendency  to  bend  the  shaft 
might  be  greater  than  the  twisting  effect. 

The  methods  of  calculating  the  size  of  shafts  for 
transmitting  a  given  power,  so  as  to  take  into 
account  both  the  twisting  and  bending  effects  pro- 
duced by  the  pull  of  the  belt  are  quite  complicated, 
and  the  beginner  will  ordinarily  find  it  best  to  use 
some  of  the  empirical  formulas  for  that  purpose 
which  are  intended  to  take  into  account  both  of 
these  effects. 

The  following  rules  by  Thurston  are  considered 
to  afford  ample  margin  for  strength  for  shafts 
which  are  well  supported  against  springing: 

To  find  the  diameter  of  a  cold  rolled  iron  shaft  to 
transmit  a  given  horse-power,  multiply  the  horse- 
power to  be  transmitted  by  75,  and  divide  the  product 
by  the  number  of  revolutions  per  minute  that  the 
shaft  is  to  make.  The  cube  root  of  this  quotient  will 
be  the  diameter  of  the  shaft. 

If  the  shaft  is  to  be  of  turned  iron,  proceed  as 
above,  except  that  the  horse-power  to  be  trans- 
mitted is  to  be  multiplied  by  125  instead  of  75. 

This  rule  is  "for  head  shafts,  supported  by  bear- 


SHAFTS,  BELTS  AND  PULLEYS       275 

ings  close  to  each  side  of  the  main  pulley  or  gear,  so 
as  to  wholly  guard  against  transverse  strain. "  If 
the  main  pulley  is  at  a  distance  from  the  bearing, 
the  size  of  the  shaft  will  need  to  be  increased, 
while  for  ordinary  line  shafting,  with  hangers  8 
feet  apart,  the  size  may  be  reduced,  figures  of  90 
for  turned  iron,  and  55  for  cold  rolled  iron  shafting 
being  substituted  for  those  given  in  the  rule ;  or,  in 
the  case  of  shafting  for  transmission  only,  without 
pulleys,  figures  of  62.5  for  turned  iron,  and  35  for 
cold  rolled  iron  are  substituted. 

To  find  the  horse-power  which  a  given  shaft  will 
transmit,  multiply  the  cube  of  its  diameter  by  the 
number  of  revolutions  per  minute,  and  divide  the 
product  by  125  for  turned  iron,  or  by  75  for  cold 
rolled  iron. 

For  line  shafting  substitute  the  figures  given  by 
90  and  55,  respectively. 

The  horse-power  which  is  being  transmitted  is 
determined  by  multiplying  the  pull  in  pounds 
which  the  belt  exerts  (or  the  push  which  the  teeth 
of  the  driving  gear  exert,  if  gears  are  used)  by 
the  diameter  of  the  pulley  in  inches  (or  the  pitch 
diameter  of  the  gear  in  inches)  and  multiplying 
this  product,  again,  by  the  number  of  revolutions 
per  minute  of  the  shaft;  then  divide  this  product 
by  126,050,  and  the  quotient  gives  the  horse-power 
transmitted. 

Expressed  as  a  formula  this  rule  would  be: 

n          P  X  D  X  N 
n'^'  126,050 


276         SELF-TAUGHT  MECHANICAL  DRAWING 

in  which  P  =  pull  on  belt  or  push  on  gear  teeth  in 
pounds, 

D  =  diameter  of  pulley  or  pitch  diameter 
of  gear  in  inches, 

N  =  number  of  revolutions  per  minute  of 
pulley  or  gear. 

Belts. — The  theoretical  horse-power  which  a  belt 
will  transmit  is  equal  to  the  pull  which  the  belt 
exerts  in  pounds,  multiplied  by  its  velocity  in  feet 
per  minute,  and  this  product  divided  by  33,000. 
The  question  then  arises  as  to  what  is  the  allowable 
stress  to  be  put  upon  a  belt. 

A  common  rule  of  practice  for  ordinary  belting 
is  that  for  single  thickness  belts  the  horse-power 
transmitted  equals  the  breadth  of  the  belt  in  inches, 
multiplied  by  its  velocity  in  feet  per  minute,  this 
product  being  divided  by  1,000.  This  rule  assumes 
a  belt  pull  of  33  pounds  per  inch  of  width.  Many 
authorities,  however,  would  allow  a  much  higher 
tension.  The  higher  the  tension,  however,  the 
narrower  the  belt  for  a  given  horse-power,  and 
the  greater  the  stretch,  the  more  frequent  the 
necessity  for  relacing,  and  the  shorter  the  life  of 
the  belt. 

Allowing  33  pounds  tension  per  inch  in  width  for 
the  thinnest  commercial  single  belt,  and  allowing 
the  tensions  for  increased  thicknesses  given  by  a 
large  belt  manufacturing  concern,  would  give  the 
following  formulas  for  the  transmission  capacities 
of  given  belts: 


SHAFTS,    BELTS  AND  PULLEYS  277 

Single  belt,       A  inch  thick,  H.P.  =  Breadth^velocity. 

Single  belt,        J  inch  thick,  H.P.  =  Breadth  X  velocity. 

800 

Light  double,    #  inch  thick,  H.P.  =  Breadth  X  velocity, 

733 

Heavy  double,  &  inch  thick,  #. P.  =  Breadth  X  velocity. 

687 

Heavy  double,  A  inch  thick,  H.P.  =  Breadth  X  velocity. 

660 

Heavy  double,    |  inch  thick,  H.P.  =Breadth  X  velocity. 

550 

Heavy  double,  if  inch  thick,  H.P.  =  Breadth  X  velocity. 

500 


In  these  formulas  the  breadth  of  the  belt  is 
understood  to  be  in  inches,  and  its  velocity  in  feet 
per  minute,  the  letters  H.P.  meaning  horse-power. 
Transposing  the  above  formulas  to  ascertain  the 
breadth  of  belt  required  to  transmit  a  given  power, 
we  would  have : 

Single  belt,       T\  inch  thick,  Breadth  =  H'  R  x  100° 

Velocity 

Single  belt,       i  inch  thick,  Breadth  =.H-P-  x  80° 

Velocity 

Light  double,  U  inch  thick,  Breadth  =  H"  P'  x  73B 

Velocity 

Heavy  double,  &  inch  thick,  Breadth  -   H'  P'  x  687 

Velocity 

Heavy  double,  T\  inch  thick,  Breadth  =  H'  P'  x  66° 

Velocity 

Heavy  double,  |  inch  thick,  Breadth  =-H'P-  x  55° 

Velocity 

Heavy  double,  ifinch  thick,  Breadth  ^  H-  P'  x  50Q 

Velocity 


278         SELF-TAUGHT  MECHANICAL  DRAWING 

These  formulas  are  all  for  laced  belts.  A  belt 
made  endless  by  being  lapped  and  cemented  or 
riveted  is  considered  to  be  nearly  50  per  cent, 
stronger  than  a  laced  belt,  and  is  thus  capable  of 
transmitting  nearly  50  per  cent,  more  power;  or 
the  breadth  of  an  endless  belt  to  transmit  a  given 
power  would  not  need  to  be  more  than  between 
two- thirds  to  three-quarters  of  the  breadth  of  a 
laced  belt.  Metal  fastenings  are  not  considered  to 
make  as  strong  a  belt  as  lacings. 

If  the  foregoing  formulas  had  been  made  on  the 
basis  of  an  allowable  stress  of  45  pounds  for  each 
inch  in  width  of  a  single  belt,  a  figure  which  many 
consider  perfectly  safe  for  a  belt  in  good  condition, 
they  would  have  shown  the  belts  as  being  capable 
of  transmitting  one-third  more  power  than  at  33 
pounds  stress  per  inch;  to  transmit  a  given  power 
a  belt  would  then  need  to  be  not  more  than  three- 
quarters  of  the  width. 

It  will  be  seen  from  these  formulas  that  the 
power  transmitting  capacity  of  a  belt  depends  upon 
its  breadth  (a  wide  belt  allowing  an  increased 
tension)  or  on  its  velocity.  Increasing  the  width 
of  the  belt  without  increasing  the  tension  to  corre- 
spond would  not  give  any  increase  of  power  trans- 
mitting capacity,  as  the  given  tension  would  simply 
be  distributed  over  so  much  more  pulley  surface ; 
but  a  tight  belt  means  more  side  strain  on  shaft 
and  journal.  Therefore,  according  to  Griffin,  from 
the  standpoint  of  efficiency,  use  a  narrow  belt  under 
low  tension  at  as  high  a  speed  as  possible.  The  de- 
sired high  speed  is,  of  course,  secured  by  simply 
putting  on  large  pulleys. 


SHAFTS,  BELTS  AND  PULLEYS       279 

Speed  of  Belting. — The  most  economical  speed 
is  somewhere  between  4000  and  5000  feet  per 
minute.  Above  these  values  the  life  of  the  belt  is 
shortened ; ' '  flapping, "  ' '  chasing, ' '  and  centrifugal 
force  also  cause  considerable  loss  of  power  at  higher 
speeds.  The  limit  of  speed  with  cast  iron  pulleys 
is  fixed  at  the  safe  limit  for  the  bursting  of  the 
rim,  which  may  be  taken  at  one  mile  surface  speed 
per  minute. 

The  formulas  given  for  the  horse-power  trans- 
mitted assume  that  the  belt  is  in  contact  with  just 
one-half  of  the  pulley ;  or,  in  other  words,  that  the 
arc  of  contact  is  180  degrees.  If  the  arc  of  contact 
is  increased,  as  it  might  be  in  the  case  of  a  crossed 
belt,  until  it  becomes  240  degrees,  or  two- thirds  of 
the  circumference  of  the  pulley,  it  is  stated  that 
the  adhesion  of  the  belt  to  the  pulley,  and  conse- 
quently the  efficiency  of  the  belt,  will  be  increased 
50  per  cent.  If,  on  the  other  hand,  the  arc  of  con- 
tact should  be  reduced  to  120  degrees,  or  one- third 
of  the  circumference  of  the  pulley,  as  might  be  the 
case  with  open  belts  where  the  shafts  were  near 
together,  and  the  pulleys  were  very  unequal  in 
size,  the  efficiency  is  stated  to  be  only  60  per  cent, 
of  what  the  formulas  would  show ;  if  the  arc  of  con- 
tact should  be  reduced  to  90  degrees,  the  efficiency 
is  stated  to  be  only  30  per  cent. 

From  these  percentages  one  can  form  a  fairly 
good  idea  of  what  percentage  to  allow  for  varying 
arcs  of  contact.  In  most  cases,  however,  it  will 
probably  be  correct  enough  to  assume  the  arc  of 
contact  to  be  180  degrees. 

In  all  cases  of  open  horizontal  belting  the  lower 


280         SELF-TAUGHT  MECHANICAL  DRAWING 

run  of  the  belt  should  be  made  the  working  part, 
so  that  the  sag  of  the  upper  run  will  increase  the 
arc  of  contact. 

In  the  location  of  shafts  that  are  to  be  connected 
with  each  other  by  belts,  care  should  be  taken  to 
secure  them  at  a  proper  distance  from  one  another. 
It  is  not  easy  to  give  a  definite  rule  what  this  dis- 
tance should  be.  Some  authorities  give  this  rule: 
Let  the  distance  between  the  shafts  be  ten  times  the 
diameter  of  the  smaller  pulley ;  but  while  this  is 
correct  for  some  cases,  there  are  many  other  cases 
in  which  it  is  not  correct.  Circumstances  generally 
have  much  to  do  with  the  arrangement;  and  the 
engineer  or  machinist  must  use  his  judgment,  mak- 
ing all  things  conform,  as  far  as  may  be,  to  general 
principles.  The  distance  should  be  such  as  to  allow 
a  gentle  sag  to  the  belt  when  in  motion.  The  Page 
Belting  Co.  states  that  if  too  great  a  distance  is 
attempted,  the  weight  of  the  belt  will  produce  a 
very  heavy  sag,  drawing  so  hard  upon  the  shafts 
as  to  produce  considerable  friction  in  the  bearings, 
while  at  the  same  time  the  belt  will  have  an  un- 
steady, flapping  motion,  which  will  destroy  both 
the  belt  and  the  machinery. 

As  belts  increase  in  width  they  should  be  made 
thicker.  It  is  advisable  to  use  double  belts  on 
pulleys  12  inches  in  diameter  and  larger.  If  thin 
belts  are  used  at  very  high  speed,  or  if  wide  belts 
are  thin,  they  almost  invariably  run  in  waves  on 
the  slack  side,  or  "travel"  from  side  to  side  of  the 
pulleys,  especially  if  the  load  changes  suddenly. 
This  waving  and  snapping  that  occurs  as  the  belts 
straighten  out,  wears  the  belts  very  fast,  and 


SHAFTS,  BELTS  AND  PULLEYS       281 

frequently  causes  the  splices  to  part  in  a  very  short 
time,  all  of  which  is  avoided  by  the  employment  of 
suitable  thickness  in  the  belts.  The  Page  Belting 
Co.  states  that  driving  pulleys  on  which  are  to  be 
run  shifting  belts  should  have  a  perfectly  flat  sur- 
face. All  other  pulleys  should  have  a  convexity  in 
the  proportion  of  about  -fj  of  an  inch  to  one  foot 
in  width.  The  pulleys  should  be  a  little  wider  than 
the  belt  required  for  the  work. 

Pulley  Sizes. — The  sizes  of  pulleys  to  give  a  re- 
quired speed,  or  the  speed  which  will  be  obtained 
with  given  pulleys  may  be  readily  found  from  the 
fact  that  the  product  of  the  speed  of  the  driving 
shaft,  in  revolutions  per  minute,  and  the  diameters 
of  all  driving  pulleys,  on  the  main  and  on  counter- 
shafts, multiplied  together,  will  be  equal  to  the 
product  of  the  diameters  of  all  driven  pulleys  and 
the  speed  of  the  last  driven  shaft,  in  revolutions 
per  minute,  multiplied  together;  so  that  if  the  size 
of  one  driven  pulley,  for  instance,  is  required,  its 
size  may  be  found  by  dividing  the  product  of  the 
speed  of  the  driving  shaft  and  all  driving  pulleys 
multiplied  together,  by  the  product  of  speed  of  the 
final  driven  shaft  and  the  diameters  of  such  driven 
pulleys  as  are  given,  multiplied  together.  The  re- 
sult will  be  the  required  pulley  size. 

Example. — A  shaft  making  200  revolutions  per 
minute  has  mounted  on  it  a  pulley  18  inches  in 
diameter  which  belts  onto  a  6-inch  pulley  on  a 
countershaft.  The  countershaft  has  mounted  on  it 
a  20-inch  pulley  which  belts  to  a  pulley  on  the 
spindle  of  a  machine  which  is  to  make  3000  revolu- 
tions per  minute.  What  size  pulley  will  be  required 
on  the  spindle. 


CAH? 


rrv 


282.       SELF-TAUGHT  MECHANICAL  DRAWING 

Placing  the  speed  of  the  driving  shaft,  and  the 
sizes  of  all  driving  pulleys  on  one  side  of  a  vertical 
line,  for  convenience  sake,  and  the  sizes  of  all 
driven  pulleys  and  the  speed  of  the  last  driven 
shaft  (or  spindle)  on  the  other  side,  and  letting  x 
represent  the  required  size  we  would  have: 


Speed  of  shaft  =  200 

Pulley  on  shaft  =    18 
Driving  pulley  on 

countershaft  =    20 


6  =  Driven  pulley  on  coun- 
tershaft. 

x  =  Required  size  of  pulley 
on  spindle. 


3000  =  Speed  of  spindle. 

i 

Then  200  X  18  X  20  =  6  X  x  X  3000 

=  200J<_18_X_20  _    _721000 
6  X~3000  18,000 

The  diameter  of  the  pulley  on  the  spindle  would 
therefore  have  to  be  4  inches.  If  this  size  had 
been  given,  and  the  speed  of  the  spindle  had  been 
required,  x  might  have  been  taken  to  represent  the 
required  speed,  when  the  same  process  would  have 
given  the  desired  information. 

Twisted  and  Unusual  Cases  of  Belting.  —  It 
frequently  happens  that,  in  transmitting  power, 
conditions  present  themselves  in  which  ordinary 
straight  belting,  either  open  or  crossed,  will  not 
serve  the  purpose,  and  recourse  must  be  had  to 
some  form  of  twisted  belting,  either  quarter  turn 
belting  or  belting  guided  by  idler  pulleys.  In  the 
following  are  given  some  of  the  principal  con- 
ditions. 

Fig.  198  shows  a  quarter  turn  belt,  by  which 
power  can  be  transmitted  from  one  shaft  to  another 
at  right  angles  to  it.  The  condition  necessary  for 


SHAFTS,  BELTS  AND  PULLEYS 


283 


the  successful  working  of  this  arrangement  is  that 
the  middle  of  the  face  of  the  pulley  toward  which 


FIG.  198.— Arrangement  of 
Pulleys  for  Quarter-Turn 
Belt. 


FIG.  199. — Another  Arrange- 
ment for  Transmitting 
Power  between  Shafts  at 
Right  Angles. 


the  belt  is  advancing  shall  be  in  line  with  the  edge 
of  the  pulley  that  the  belt  is  leaving.    An  exami- 


284        SELF-TAUGHT  MECHANICAL  DRAWING 

nation  of  both  the  plan  and  elevation  views  will 
make  this  clear. 

While  this  is  the  simplest  arrangement  for  this 
purpose,  it  has  several  drawbacks.  The  edgewise 
stress  on  the  belt  as  it  is  leaving  either  pulley  is 
very  severe  on  the  belt.  It  also  causes  a  consider- 
able loss  of  contact  with  the  pulley  face,  with 
corresponding  loss  of  power  transmission  capacity. 
The  edgewise  stress  also  makes  it  necessary,  if 
durability  is  to  be  considered,  to  have  the  belt 
relatively  narrow.  Incidentally,  also,  any  reversal 
of  the  motion  will  cause  the  belt  to  immediately 
run  off  the  pulleys. 

Fig.  199  shows  another  arrangement  for  trans- 
mitting power  from  one  shaft  to  another  at  right 
angles  to  it,  which  overcomes  all  of  the  objections 
mentioned  to  the  arrangement  shown  in  Fig.  198, 
but  at  the  expense  of  a  double  length  belt  and  an 
extra  pair  of  pulleys. 

As  shown  in  the  illustration,  A  and  B  are  tight 
pulleys,  and  C  and  D  are  loose  pulleys.  The  belt, 
as  it  leaves  the  tight  pulley  A,  passes  down  under 
the  loose  pulley  D,  up  over  the  loose  pulley  C,  down 
under  the  tight  pulley  B,  and  then  up  over  the 
tight  pulley  A,  making  a  complete  circuit.  The 
loose  pulleys,  it  will  be  seen,  revolve  in  an  opposite 
direction  to  the  shafts  on  which  they  are  mounted. 

Fig.  200  shows  an  arrangement  by  which,  by 
employing  loose  guide  pulleys,  power  may  be  trans- 
mitted from  one  shaft  to  another  so  close  to  it  as 
to  prohibit  direct  belting.  If  the  main  pulleys  are 
of  the  same  size,  and  their  shafts  are  in  the  same 
plane,  the  guide  pulleys  may  be  mounted  on  a 


SHAFTS,  BELTS  AND  PULLEYS       285 

single  straight  shaft  at  right  angles  to  a  plane 
passing  through  the  axes  of  the  shafts  on  which 
the  main  pulleys  are  mounted.  If,  however,  the 
main  pulleys  are  of  unequal  size,  as  shown  in  the 
illustration,  the  guide  pulleys  will  have  to  be  in- 
clined to  such  an  angle  that  the  center  of  the  face 


FIG.  200.  —Arrangement  of  Belt  Transmission  Using 
Loose  Guide  Pulleys. 

of  the  pulley  toward  which  the  belt  is  advancing 
shall  be  in  line  with  the  edge  of  the  pulley  that 
the  belt  is  leaving,  the  same  as  in  the  case  of  the 
quarter  turn  belt  shown  in  Fig.  198. 

It  is  not  necessary  that  the  shafts  on  which  the 
main  pulleys  are  mounted  be  in  the  same  plane ; 
their  direction  may  be  such  that  their  relation  to 


286         SELF-TAUGHT  MECHANICAL  DRAWING 

each  other  is  similar  to  that  of  those  shown  in 
Fig.  198,  or  at  any  intermediate  angle. 
Again,  if  they  are  in  the  same  plane,  it  is  not 


necessary  that  they  should  be  parallel  with  each 

other;  they  may  be  at  any  angle  with  each  other. 

Fig.  201  shows  a  case  which  is  a  modification  of 

Fig.  200.     The  main  shafts  are  at  right  angles  to 


SHAFTS,  BELTS  AND  PULLEYS 


287 


each  other.  The  main  pulleys,  being  of  the  same 
size,  permit  the  guide  pulleys  to  be  mounted  on 
a  single  shaft.  This  arrangement  is  a  common 
method  of  transmitting  power  around  a  corner. 

Fig.  202  shows  a  case  where  the  direction  of  the 
shafts  with  regard  to  each  other  is  the  same  as  in 


FIG.  202.— A  Case  where  the  Guide  Pulleys  would  be  Mounted 
in  an  Adjustable  Frame. 

Fig.  198,  but  where  shop  conditions  are  such  that 
it  is  not  practicable  to  bring  the  lower  shaft  under 
the  upper  one  to  permit  of  belting  by  either  of  the 
methods  shown  in  Figs.  198  or  199.  The  guide 
pulleys  are,  therefore,  mounted  on  a  frame  which 
can  be  raised  or  lowered  in  guides  by  means  of  an 
adjusting  screw,  permitting  of  an  easy  adjustment 
of  the  belt  tension. 


288         SELF-TAUGHT  MECHANICAL  DRAWING 


Fig.  203  shows  a  case  which  is  similar  to  Fig. 
200  in  that  it  permits  the  belting  together  of  shafts 
which  are  at  angle  to  each 
other,  but  accomplishes  this 
result  by  the  use  of  only  one 
guide  pulley.  The  shafts, 
though  at  an  angle  to  each 
other,  are  in  the  same  plane. 
This,  however,  is  not  neces- 
sarily so.  The  shafts  may 
be  twisted  around  until  they 
are  at  right  angles  to  each 
other,  as  in  Fig.  198.  As 
shown  in  Fig.  200,  the  belt 
may  be  run  in  either  direc- 
tion as  long  as  the  shafts  are 
in  the  same  plane;  but  as 
shown  in  Fig.  203,  it  is  nec- 
essary that  the  belt  should 
be  run  in  the  direction  in- 
dicated by  the  arrows. 

An  examination  of  the  en- 
gravings will  show  that  the 
condition  necessary  for  the 
proper  working  of  guide 
pulleys  is  that  the  shaft  on 
which  the  guide  pulley  is  mounted  shall  be  at  right 
angles  to  a  line  drawn  from  the  edge  of  the  pulley 
that  the  belt  is  leaving  in  its  advance  toward  the 
guide  pulley,  to  the  middle  of  the  guide  pulley 
face. 


FIG.  203.— An  Arrange- 
ment in  Which  but  One 
Guide  Pulley  is  Used. 


CHAPTER  XVIII 

FLY-WHEELS  FOR  PRESSES,  PUNCHES,  ETC. 

IN  a  great  many  different  classes  of  machinery, 
the  work  that  the  machine  performs  is  of  a  variable 
or  intermittent  nature,  being  done,  in  the  case,  for 
example,  of  punches  and  presses,  during  a  small 
part  of  the  time  required  for  the  driving  shaft  or 
spindle  of  the  machine  to  make  a  complete  revolu- 
tion. If  this  work  could  be  distributed  over  the 
entire  period  of  the  revolution,  a  comparatively  nar- 
row belt  would  be  sufficient  to  drive  the  machine; 
but  a  very  broad  and  heavy  belt  would  otherwise  be 
necessary  to  overcome  the  resistance,  if  the  belt 
only  be  depended  on  to  do  the  work.  It  is,  of 
course,  in  a  sense,  impossible  to  distribute  the  work 
of  the  machine  over  the  entire  period  of  revolution 
of  the  driving  shaft  of  the  machine,  but  by  placing 
a  large,  heavy-rimmed  wheel,  a  fly-wheel,  on  the 
shaft,  the  belt  is  given  an  opportunity  to  perform 
an  almost  uniform  amount  of  work  during  the 
whole  revolution.  During  the  greater  part  of  the 
revolution  of  the  driving  shaft  the  power  of  the 
belt  is  devoted  to  accelerating  the  speed  of  the  fly- 
wheel. During  that  brief  period  of  the  revolution 
of  the  shaft  when  the  work  of  the  machine  is  being 
done,  the  energy  thus  stored  up  in  the  fly-wheel  is 
given  out  at  the  expense  of  its  velocity.  The 

289 


290         SELF-TAUGHT  MECHANICAL  DRAWING 

energy  a  fly-wheel  would  give  out  if  brought  to  a 
standstill  would  be  (neglecting  the  weight  of  the 
arms  and  hub,  as  the  efficiency  of  the  wheel  depends 
chiefly  on  the  weight  of  the  rim),  expressed  in 
foot-pounds,  equal  to  the  weight  of  the  rim  in 
pounds  multiplied  by  the  square  of  its  velocity  at 
its  mean  diameter  in  feet  per  second,  and  this 
product  divided  by  64.32,  the  same  as  in  the  case 
of  a  falling  body  moving  at  the  same  velocity,  as 
explained  in  the  section  on  mechanics. 
Expressed  as  a  formula  this  rule  is : 

=  Wv*       Wv2 
2g         64.32 

in  which  E  =  total  energy  of  fly-wheel, 

W  =  weight  of  fly-wheel  rim  in  pounds, 
v  =  velocity  at  mean  radius  of  fly-wheel 

in  feet  per  second, 
g  =  acceleration  due  to  gravity  =  32. 16. 

If  the  speed  of  the  fly-wheel  is  only  reduced,  the 
energy  which  it  would  give  out  would  be  equal  to 
the  difference  between  the  energy  which  it  would 
give  out  if  brought  to  a  full  stop,  and  that  which 
it  would  still  have  stored  up  in  it  at  its  reduced 
velocity.  Therefore,  to  find  the  energy  in  foot- 
pounds which  a  fly-wheel  will  give  out  with  an 
allowable  loss  of  speed,  subtract  the  square  of  the 
velocity  of  the  rim  in  feet  per  second  at  its  reduced 
speed  from  the  square  of  its  velocity  in  feet  per 
second  at  full  speed,  multiply  this  difference  by 
the  weight  in  pounds,  and  divide  the  product  by 
64.32.  The  result  will  give  the  loss  of  energy  in 
foot-pounds. 


FLY-WHEELS  291 

This  long  and  cumbersome  rule  is  expressed 
much  more  simply  by  the  formula: 

ElS=       ""64732" 

in  which  El  =  energy,   in  foot-pounds,  fly-wheel 

gives  out  while  speed  is  reduced 

from  Vi  to  v2 , 
Vi  =  speed  before  any  energy  has  been 

given  out,  in  feet  per  second, 
v2  =  speed  at  end  of  period  during  which 

energy  has  been  given  out,  in  feet 

per  second, 
W  =  weight  of  fly-wheel  rim  in  pounds. 

This  rule  and  formula  may  be  transposed  as  fol- 
lows :  To  find  the  weight  of  a  fly-wheel  to  give  out 
a  required  amount  of  energy  with  an  allowable  loss 
of  speed,  multiply  the  required  amount  of  energy 
in  foot-pounds  by  64.32,  and  divide  the  product  by 
the  difference  between  the  square  of  the  velocity 
of  the  rim,  at  its  mean  diameter,  in  feet  per  second 
at  full  speed,  and  the  square  of  its  velocity  in  feet 
per  second  at  its  reduced  speed ;  or,  expressed  as  a 
formula,  using  the  same  notation  as  above : 

v  i2  —  v  ? 

When  the  mean  diameter  of  the  fly-wheel  is 
known,  the  velocity  of  the  rim  at  its  mean  diameter 
in  feet  per  second  will  be 

Diameter  in  feet  X  3.1416  X  rev,  per  minute 
~60 

It  is  evident  that  in  designing  a  fly-wheel  for  a 


292         SELF-TAUGHT   MECHANICAL  DRAWING 

machine,  there  is  an  opportunity  for  a  wide  range 
in  the  weight,  from  a  wheel  heavy  enough,  when 
once  it  has  been  brought  to  its  full  speed,  to  do,  by 
means  of  the  energy  stored  in  it,  the  work  without 
assistance  from  the  belt,  the  belt  being  only  just 
wide  enough  to  restore  the  speed  of  the  wheel  in 
time  for  the  next  operation,  to  a  wheel  where  the 
belt  is  wide  enough  to  do  the  most  of  the  work 
directly,  the  stored  energy  in  the  fly-wheel  merely 
assisting  it  somewhat.  Perhaps  the  best  way  would 
be  to  have  the  wheel  heavy  enough  so  that  its 
stored  energy  could  do  the  bulk  of  the  work,  the 
belt  assisting  it,  and  at  the  same  time  have  the 
latter  wide  enough  to  quickly  restore  the  speed  of 
the  wheel,  so  that,  in  case  its  velocity  should  be 
reduced  beyond  that  calculated,  there  would  be  a 
margin  of  available  power  in  the  belt. 

Example. — Let  it  be  required  to  design  a  fly- 
wheel for  a  press  to  cut  off  one-inch  round  bar 
steel,  the  press  making  30  strokes  per  minute. 

Soft  steel  having  a  shearing  resistance  of  about 
50,000  pounds  per  square  inch,  and  a  one-inch  bar 
having  an  area  of  cross-section  of  0.7854  square 
inch,  the  shearing  resistance  of  the  bar  will  be 
50,000  X  0.7854  =  39,270  pounds,  or  practically 
40,000  pounds.  This  resistance  varies,,  however, 
during  the  process  of  shearing,  being  greatest  near 
the  beginning  of  the  cut,  and  decreasing  as  the 
cutting  progresses.  In  the  case  of  a  round  bar  it 
could  not  decrease  uniformly,  because  of  the  shape 
of  the  cross-section.  For  the  sake  of  getting  the 
decrease  in  resistance  as  nearly  uniform  as  possible, 
we  will  assume  that  the  work  of  cutting  off  a  one- 


FLY-WHEELS  293 

inch  round  bar  is  the  same  as  the  work  of  cutting 
off  a  square  bar  of  the  same  area ;  though  this  may 
not  be  quite  exact,  it  would  probably  not  be  far 
out  of  the  way.  The  length  of  the  sides  of  a  square 
of  the  same  area  as  a  given  circle,  is  equal  to  the 
diameter  of  the  circle  multiplied  by  0.886.  There- 
fore, our  equivalent  square  bar  will  be  0.886  of 
an  inch  square.  The  mean  resistance  to  cutting, 
assuming  that  the  resistance  decreases  uniformly 
as  the  cutting  progresses,  would  be  40,000  ^  2  = 
20,000  pounds.  As  the  cutting  operation  continues 
through  a  space  of  0.886  of  an  inch,  the  power 
required  would  be  20,000  X  0.886  =  17,720  inch- 
pounds,  or  1476.6  foot-pounds.  Let  us  plan  to  have 
the  belt  do  one-fifth  of  the  work  of  cutting  direct- 
ly, leaving  four-fifths  to  be  done  by  the  stored  up 
energy  of  the  fly-wheel.  One-fifth  of  1476.6  equals 
295.3.  Subtracting  this  from  1476.6  leaves  1181.3 
foot-pounds  to  be  supplied  by  the  energy  of  the 
fly-wheel.  As  a  preliminary  calculation  let  us  find 
what  would  have  to  be  the  weight  of  the  wheel  if 
it  were  to  be  placed  upon  the  crank-shaft,  the  shaft 
which  operates  the  plunger  of  the  press.  Assuming 
the  mean  diameter  of  the  fly-wheel  rim  to  be  4 
feet,  the  circumference  would  be  4  X  3.14  =  12.56 
feet,  and,  as  the  shaft  makes  30  revolutions  per 
minute,  the  velocity  of  the  rim  in  feet  per  second 
would  be : 

12.56  X  30       c  00  , 
— gg =  6.28  feet. 

If  we  expect  the  fly-wheel  to  suffer  a  loss  of, 
say,  10  per  cent,  while  doing  its  work,  then  its 
velocity  at  its  reduced  speed  will  be  6.28  -  0.628  = 


294        SELF-TAUGHT  MECHANICAL  DRAWING 

5.65  feet.  The  weight  of  the  fly-wheel  to  give  out 
1181. 3  foot-pounds  under  these  conditions  will  then 
be,  according  to  the  rule  and  formula  already 
given : 

1181.3  X  64.32_      75,981.2        75.981.2  _  1  n 
6.282-5.652      39.44-31.92  =     7.52 
nearly. 

A  wheel  weighing  10, 100  pounds  would,  of  course, 
be  out  of  the  question ;  but  as  the  energy  increases 
as  the  square  of  the  velocity,  the  weight  may  be 
very  rapidly  reduced  by  mounting  the  wheel  upon 
a  higher-speeded  secondary  shaft,  connected  with 
the  crank-shaft  by  reducing  gears.  If  the  speed 
of  the  secondary  shaft  is  to  the  speed  of  the  crank- 
shaft as  6  to  1,  the  weight  of  the  wheel,  if  the 
mean  diameter  be  kept  the  same,  will  need  to  be 
only  about  one  thirty-sixth  of  what  it  would  need 
to  be  if  mounted  on  the  crank-shaft.  At  thisjhigher 
speed,  however,  it  might  be  desirable  to  somewhat 
reduce  the  diameter  of  the  wheel.  Let  us  assume 
that  the  mean  diameter  be  made  3  feet.  If  the 
ratio  of  speeds  is  6  to  1,  the  wheel  will  make  180 
revolutions  per  minute,  and  the  velocity  of  the  rim 
in  feet  per  second  will  be : 

3  X  3.14  X  180  '        0  , 
To"          =28-3  feet. 

If  the  wheel  suffers  a  loss  of  10  per  cent.,  its 
velocity  at  its  reduced  speed  will  be  28.3  -  2.83  = 
25.5  nearly.  The  weight  of  the  wheel  will  then 
be: 

1181.3  X  64.32       75,981.2       Kn A  , 

28.3 2- 25.5 2    =  15064^  =  5°4  P°Unds' 


FLY-WHEELS  295 

As  a  cubic  inch  of  cast  iron  weighs  0.26  pound, 
the  wheel  will  contain  .  504  -5-  0.26  =  1938  cubic 
inches.  The  mean  circumference  of  the  rim  in 
inches  will  be  3  X  12  X  3.14  =  113  inches.  The 
cross-section  of  the  rim  will  then  be : 

1938  -  113  =  17.1  square  inches. 

This  would  mean  a  rim  about  4  by  4J  inches. 
The  outside  diameter  of  the  wheel  would  then  be 
40  inches. 

We  planned  to  have  the  belt  do  one-fifth  of  the 
work,  and  this  we  found  to  be  295.3  foot-pounds. 
If  the  crank  has  a  radius  of  li  inch,  the  cutter  will 
have  a  stroke  of  2J  inches,  and  if  the  cutters  over- 
lap each  other  one-quarter  of  an  inch  at  the  end  of 
the  stroke,  the  crank  will  have  to  swing  through 
an  angle  of  about  54  degrees  in  order  to  make  the 
cutters  advance  the  one  inch  necessary  to  cut  off 
the  one-inch  bar,  as  a  simple  lay-out  will  show. 
The  belt  must  then  develop  295.3  foot-pounds  while 
the  crank  swings  through  54  degrees.  It  will  then 
develop  295.3  +  54  =  5.5  foot-pounds,  nearly,  in  one 
degree,  and  in  a  complete  revolution  it  will  develop 
5.5  X  360  =  1980  foot-pounds.  As  the  press  makes 
30  strokes  per  minute,  the  belt  will  develop  30  X 
1980  =  59,400  foot-pounds  per  minute.  If  a  driving 
pulley  18  inches  in  diameter  is  used,  the  belt  speed 
in  feet  per  minute  will  be : 

18  X  3.14  X  180  a  , 

— ^—         =  848  feet. 

If  a  single  thickness  belt,  one-inch  wide,  at 
1000  feet  per  minute,  transmits  33,000  foot-pounds 


296         SELF-TAUGHT  MECHANICAL  DRAWING 

per  minute,  the  same  belt  at  848  feet  per  minute 
will  transmit  TWo  as  much,  or  33,000X0.848  = 
27,984  foot-pounds.  The  width  of  belt  necessary 
to  transmit  59,400  foot-pounds  per  minute  at  this 
speed  will  then  be  59,400  -*-  27,984  =  2.1  inches. 
No  account  has  so  far  been  taken  of  the  power 
necessary  to  drive  the  machine  itself.  To  allow 
for  this  the  belt  should  evidently  be  not  less  than 
2i  inches  wide.  A  3-inch  belt  would  allow  consid- 
erable of  a  margin  of  safety,  and  further  calculation 
will  show  that  such  a  belt  would  develop,  during 
about  one- third  of  a  revolution  of  the  crank,  the 
amount  of  energy  which  the  fly-wheel  had  lost,  so 
that,  as  the  cutting  operation  takes  about  one-sixth 
of  a  revolution,  the  fly-wheel  would  be  running  at 
full  speed  for  about  one-half  of  a  revolution  of  the 
crank,  previous  to  the  beginning  of  the  cut,  pro- 
vided that  it  had  not  suffered  any  greater  reduction 
of  velocity  than  the  10  per  cent,  planned  for. 

If  the  press  was  employed  doing  punching  the 
same  method  of  procedure  would  be  employed  in 
the  calculations,  the  area  in  shear  in  such  a  case 
being  equal  to  the  circumference  of  the  hole  mul- 
tiplied by  the  thickness  of  the  plate.  The  end  of  a 
punch .  is  usually  made  slightly  conical  or  slightly 
beveling,  the  effect  in  either  case  being  to  increase 
the  shearing  action,  and  make  the  work  of  punch- 
ing easier. 


CHAPTER  XIX 

TRAINS  OF  MECHANISM 

FOR  obtaining  high  speeds  without  the  use  of 
unduly  large  driving  pulleys  or  gears,  for  securing 
gain  in  power  by  sacrificing  speed,  for  securing 
reversal  of  direction,  or  for  obtaining  some  par- 
ticular velocity  ratio  between  the  driver  and  some 
part  of  the  mechanism,  pulleys,  gears,  worm-gears, 
or  the  like,  may  be  substituted  for  direct  acting 
driving-mechanisms. 

To  Secure  Increase  of  Speed. — Let  a  shaft  making 
100  revolutions  per  minute  be  required  to  drive  the 
spindle  of  a  machine  at  2000  revolutions  per  minute, 
the  pulley  on  the  spindle  being  3  inches  in  diam- 
eter. If  a  direct  drive  were  to  be  used,  the  pulley 
on  the  shaft  would  have  to  be  as  many  times  greater 
than  the  pulley  on  the  spindle  as  2000  is  greater 
than  100,  or  20  times. 

This  would  mean  a  pulley  on  the  shaft  60  inches 
in  diameter.  Practical  considerations,  such  as  the 
weight  of  the  pulley,  size  of  hangers  and  the  like, 
would  make  such  a  pulley  out  of  the  question. 

By  interposing  an  intermediate  countershaft  be- 
tween the  first  shaft  and  the  spindle  of  the  machine, 
however,  having  pulleys  of  such  size  that  the 
product  of  the  ratio  of  the  pulley  on  the  first  shaft 
and  the  one  to  which  it  is  belted  on  the  counter- 
shaft, multiplied  by  the  ratio  of  the  second  pulley 

297 


298         SELF-TAUGHT  MECHANICAL  DRAWING 

on  the  countershaft  and  the  pulley  on  the  spindle 
to  which  it  is  belted  is  equal  to  the  ratio  which 
it  is  desired  to  have  between  the  first  shaft  and 
the  spindle,  the  same  speed  may  be  secured  by  the 
use  of  pulleys  of  convenient  size.  Thus,  if  the 
ratio  between  the  pulley  on  the  first  shaft  and  the 
one  on  the  countershaft  is  as  1  to  4,  and  the  ratio 
between  the  driving  pulley  on  the  countershaft 
and  the  one  on  the  spindle  of  the  machine  is  as 
1  to  5,  the  product  of  these  two  ratios,  1  to  4  and  1 
to  5,  is  1  to  20,  and  the  arrangement  will  give  the 


FIG.  204.— Reversal  of  Direction  Obtained  by  Crossed  Belt. 

required  speed.  The  pulley  on  the  spindle  being  3 
inches  in  diameter,  the  driving  pulley  on  the  coun- 
tershaft will  be  15  inches  in  diameter,  and  if  the 
driven  pulley  on  the  countershaft  is  4  inches  in 
diameter  the  pulley  on  the  first  shaft  to  which  it  is 
belted  will  be  16  inches  in  diameter,  instead  of  60 
inches,  as  would  be  required  with  direct  belting. 

If  the  spindle  of  the  machine,  instead  of  being 
driven  were  made  the  driver,  as  it  would  be  if  it 
were  the  armature  shaft  of  a  motor,  then  this  ar- 
rangement would  give  gain  in  power  with  con- 
sequent loss  of  speed. 

To  Secure  Reversal  of  Direction.— In  cases  where 
shafts  are  belted  together,  reversal  of  direction  of 


TRAINS  OF  MECHANISM 


299 


rotation  is  secured  by  simply  using  a  crossed  belt 
instead  of  an  open  one,  as  shown  in  Fig.  204. 
When  gears  are  used,  reversal  of  direction  of  rota- 
tion follows  as  a  natural  condition  of  their  meshing 
together,  as  shown  in  Fig.  205.  In  order  that  the 
two  gears  A  and  B  shall  rotate  in  the  same  direc- 
tion, it  is  necessary  to  separate  them  slightly,  and 
interpose  an  intermediate  gear,  or  idler,  between 


FIG.  205.—  Relative  Direc- 
tion of  Rotation  in  a 
Pair  of  Gears. 


FlG.  206.— Influence  of  Idler 
on  Direction  of  Rotation. 


them  as  shown  in  Fig.  206.  The  rates  of  rotation 
of  A  and  B  with  regard  to  each  other  is  not  affected 
by  the  idler  gear,  whether  the  idler  be  large  or 
small.  That  this  is  so  may  be  seen  by  direct  exam- 
ination. If  A  is  the  driver,  its  circumference  will 
impart  to  the  circumference  of  C  its  own  rate  of 
motion,  and  C  will  in  turn  impart  to  B  the  same 
rate  of  motion,  which  is  the  same  as  it  would  have 
if  in  direct  connection  with  A. 

If,  now,  another  idler  be  interposed  between  A 
and  B,  making  four  gears  in  the  train,  A  and  B 
will  again  rotate  in  opposite  directions.  From  this 
it  will  be  seen  that  when  a  train  is  composed  of  an 


300         SELF-TAUGHT  MECHANICAL  DRAWING 

even  number  of  gears,  the  first  and  last  members 
rotate  in  opposite  directions ;  but  when  the  train  is 
composed  of  an  odd  number  of  gears,  the  first  and 
last  members  rotate  in  the  same  direction. 

In  Fig.  207  is  shown  the  mechanism  used  in 
engine  lathes  to  secure  either  direct  or  reversed 
motion,  by  having  the  working  train  consist  of 
either  an  even  or  an  odd  number  of  gears.  In  this 


FIG.  207.— Principle  of  Turn-      FIG.  208.— Principle  of  Com- 
bler  Gear.  pound  Idler. 

arrangement  A  is  a  gear  on  the  head-stock  spindle, 
and  B  is  a  gear  on  a  stud  below.  Pivoted  on  the 
axis  of  B  is  a  triangular  piece  of  metal,  or  bracket, 
shown  in  dotted  lines,  which  can  be  swung  back 
and  forth  by  the  handle  E.  Mounted  on  this 
bracket  are  the  idler  gears  C  and  D,  C  being  con- 
stantly in  mesh  with  B,  and  D  being  in  mesh  with 
C.  When  it  is  required  that  B  shall  rotate  in  the 
same  direction  as  A,  the  handle  E  is  lowered  until 
C  meshes  with  A.  The  working  train  then  consists 


TRAINS  OF  MECHANISM  301 

of  three  gears,  A,  C  and  B,  D  being  out  of  mesh  with 
A,  revolving  by  itself,  but  not  forming  a  part  of 
the  working  train.  When  it  is  desired  that  B  shall 
rotate  in  the  opposite  direction  to  A,  the  handle  E 
is  raised  until  D  meshes  with  A,  C  being  thrown 
out  of  mesh  with  it.  The  working  train  then  con- 
sists of  four  gears,  A,  D,  C  and  B,  and  the  desired 
reversal  is  secured. 

The  Compound  Idler. — It  has  been  shown  that 
when  a  train  consists  of  simple  gears  the  relative 
rates  of  rotation  of  the  first  and  last  members  re- 
main unchanged,  regardless  of  the  number  or  size 
of  the  idlers  that  may  be  interposed.  When  it  is 
desired  to  secure  a  different  rate  of  rotation  be- 
tween two  members  of  a  train  than  that  which 
they  would  have  if  meshing  directly  together,  a 
compound  idler  is  used,  as  shown  in  Fig.  208.  Such 
a  gear  is  used  on  many  screw  cutting  lathes.  For 
cutting  threads  up  to  a  certain  number  per  inch 
the  screw  cutting  train  consists  of  simple  gears. 

A  compound  idler  may  then  be  introduced  into 
the  train,  when  without  other  change  additional 
threads  may  be  cut.  If  with  screw  cutting  trains 
of  simple  gears  a  lathe  will  cut  all  whole  numbers 
of  threads  up  to  13  threads  per  inch,  then,  by  adding 
a  compound  idler  to  the  train,  having  its  two  steps 
in  the  ratio  of  2  to  1,  threads  from  14  to  26  per  inch 
(except  odd  numbers)  may  be  cut  with  the  same 
gears  as  previously  used  for  cutting  up  to  13  threads 
per  inch.  If  the  compound  idler  forms  an  additional 
member  of  the  train,  the  reversal  of  direction  of 
rotation  which  would  take  place  in  the  motion 
of  the  lead-screw  of  the  lathe  may  be  taken  care  of 


302        SELF-TAUGHT  MECHANICAL  DRAWING 


by  the  reversing  gears  between  the  spindle  of  the 
head-stock  and  the  stud,  previously  described,  and 
shown  in  Fig.  207. 

The  Screw  Cutting  Train.— In  Fig.  209  is  shown 
the  screw  cutting  mechanism  found  on  engine 
lathes.  The  reversing  mechanism  shown  in  Fig. 


FIG.  209.  FIG.  210. 

FIGS.  209  and  210.— Arrangement  of  Lathe  Change  Gearing. 

207  is  reproduced  entire,  and  these  gears — the  gear 
A  on  the  lathe  spindle,  the  gear  B  on  the  stud, 
which  is  connected  with  A  by  the  idlers  C  and  D— 
are  all  permanent  gears.  These  gears  are  usually 
on  the  inside  of  the  head-stock  as  shown  in  Fig. 
210.  The  stud  reaches  through  the  head-stock,  and 
on  its  outer  end  is  the  change  gear  F,  connecting 
with  the  change  gear  G  on  the  lead-screw  of  the 
lathe  by  means  of  the  intermediate  idler  H.  The 
idler  H  is  mounted  on  a  slotted  swinging  arm  as 
shown,  so  as  to  allow  of  gears  F  and  G  being 


TRAINS  OP  MECHANISM  303 

replaced  by  others  of  such  size  as  may  be  required 
to  cut  the  particular  screw  desired.  The  carriage  of 
the  lathe,  carrying  the  screw  cutting  tool,  is  driven 
directly  by  the  lead-screw.  On  large  lathes  this 
screw  is  quite  coarse,  four  threads  per  inch  being 
common,  while  on  smaller  lathes  a  finer  thread  is 
used.  The  gear  A  on  the  spindle  and  the  fixed  gear 
B  on  the  stud  are  sometimes  of  the  same  size,  and 
sometimes  of  different  sizes. 

The  problem  met  with  in  screw  cutting  is  to  find 
what  sizes  change  gears,  F  and  G,  must  be  used 
so  that  the  lead-screw  shall  drive  the  carriage  along 
one  inch  while  the  spindle  of  the  lathe  is  making 
a  number  of  revolutions  equal  to  the  number  of 
threads  to  be  cut  per  inch.  Let  us  take  as  an 
example  the  assumed  case  of  a  lathe  in  which  the 
lead-screw  has  9  threads  per  inch,  and  in  which 
the  number  of  teeth  in  the  gear  on  the  spindle  is 
to  the  number  of  teeth  in  the  fixed  gear  on  the  stud 
as  3  to  4;  required  the  size  of  change  gears  to  cut 
23  threads  per  inch.  Then,  as  the  lead-screw  has 
9  threads  per  inch,  the  spindle  of  the  lathe  must 
make  23  revolutions  while  the  lead-screw  is  making 
9  revolutions.  The  method  used  in  a  previous 
chapter  for  obtaining  the  size  of  pulleys  to  give 
required  speeds  will  give  us  the  solution  of  this 
problem;  if  the  speed  of  the  first  driving  member 
of  the  train,  together  with  the  number  of  teeth  or 
relative  sizes  of  all  other  driving  members  be  placed 
on  one  side  of  a  vertical  line,  and  the  speed  of  the 
last  driven  member,  together  with  the  number  of 
teeth  or  relative  sizes  of  all  other  driven  members 
be  placed  on  the  other  side  of  the  line,  the  product 


304         SELF-TAUGHT  MECHANICAL  DRAWING 

of  the  numbers  on  one  side  of  the  line  multiplied 
together  will  equal  the  product  of  the  numbers  on 
the  other  side  of  the  line  multiplied  together.  The 
spindle  of  the  lathe  is,  of  course,  the  first  driving 
member  of  the  train,  and  the  lead-screw  is  the 
last  driven  member.  As  the  spindle  is  to  make  23 
revolutions  while  the  lead-screw  makes  9  revolu- 
tions, 23  will  be  the  first  number  on  the  side  of  the 
line  on  which  the  driving  members  are  placed,  and 
9  will  be  the  last  number  on  the  side  of  the  line  on 
which  the  driven  members  are  placed.  Next,  as 
the  ratio  between  the  sizes  of  the  driving  gear  on 
the  lathe  spindle  and  the  fixed  gear  on  the  stud 
below  which  it  drives  is  as  3  to  4,  these  numbers 
will  be  placed  against  each  other  on  opposite  sides 
of  the  line. 

The  ratio  between  the  numbers  of  teeth  or  sizes 
of  the  two  change  gears,  F  and  G,  whose  sizes  it 
is  required  to  find,  being  unknown,  may  be  said  to 
be  as  1  to  the  unknown  number  x.  These  numbers, 
1  and  05,  are  now  placed  on  their  proper  sides  of 
the  line,  and  the  problem  appears  as  shown  below. 
The  size  of  the  idler  gear  H  does  not  enter  into  the 
question,  because,  as  has  been  previously  shown,  a 
simple  idler  gear  does  not  affect  the  relative  rates 
of  rotation  of  the  gears  between  which  it  transmits 
motion. 


Speed  of  spindle  23 

Ratio  of  size  of  spindle  gear  3 


4  to  size  of  fixed  stud  gear. 


Ratio  of  number  of  teeth 
in  change  gear  F  1   !  x  to    number    of     teeth   in 

change  gear  G 
9  speed  of  lead-screw. 


69  =  ZGx 


TRAINS  OF  MECHANISM  305 

Multiplying  together  the  numbers  on  both  sides 
of  the  line  gives  the  equation  69  =  36x.  It  is  evi- 
dent that  if  69  equals  36x,  x  must  be  equal  to  69 
divided  by  36,  or  f|.  The  ratio  between  sizes  of 
the  gear  F  and  the  gear  G  is  then  as  1  to  ||. 
Eliminating  the  fraction  by  multiplying  both  terms 
of  the  ratio  by  36  gives  the  ratio  as  36  to  69.  If, 
then,  F  has  36  teeth,  and  G  has  69  teeth,  the  lathe 
will  cut  the  required  number  of  23  threads  per 
inch. 

In  Fig.  211  is  shown  how  a  compound  idler  gear 
is  sometimes  used  in  a  screw  cutting  train.  The 


FIG.  211.— Compound  Gearing. 

change  gear  G  and  the  idler  H  have  long  hubs  on 
one  side.  When  it  is  desired  to  cut  finer  threads 
than  what  the  gears  E  and  G  with  the  idler  H  will 
give,  H  and  G,  are  put  on  with  the  long  hubs 
toward  the  lathe,  throwing  them  out  of  line  with 
E.  The  gear  E  then  meshes  into  the  large  step  of 
7,  the  small  step  of  /  meshes  into  H,  and  H  meshes 


306        SELF-TAUGHT  MECHANICAL  DRAWING 

into  G.  The  ratio  between  the  large  and  the  small 
steps  of  /  must  then  be  taken  into  account  in  the 
calculation.  For  cutting  the  coarser  threads  H  and 
G  are  put  on  with  the  short  hubs  toward  the  lathe, 
bringing  them  into  line  with  E.  The  idler  /  is  also 
turned  over,  so  that  its  large  step  is  on  the  outside 
and  out  of  line  with  E  and  H.  It  is  then  swung 
back  out  of  the  way. 

When  the  gearing  is  fully  compounded  the  two 
gears  at  /  are  separate  from  each  other  but  keyed 
together  on  the  same  stud  and  mounted  in  the 
same  manner  as  shown  in  Fig.  211.  By  varying 
the  sizes  of  these  gears,  almost  any  screw  thread 
may  be  cut  within  reasonable  limits.  In  this  case, 
of  course,  there  are  four  gears  to  be  determined  in 
our  calculations.  Simplified  rules  are  given  in  the 
following  for  this  case,  as  well  as  for  the  regular 
simple  trains. 

Large  lathes  are  provided  with  change  gears  for 
cutting  threads  from  about  2  to  about  20  threads 
per  inch,  smaller  lathes  being  provided  with  gears 
for  cutting  from  about  3  or  4  to  40  or  50  threads  per 
inch,  in  either  case  including  a  pair  of  gears  for 
cutting  11J  threads  per  inch,  this  being  the  stand- 
ard thread  for  iron  pipes  from  one  to  two-inch  sizes 
inclusive.  The  smaller  lathes  would  also  naturally 
be  provided  with  gears  for  cutting  27  threads  per 
inch,  this  being  the  number  of  threads  on  i-inch 
iron  pipes. 

Simplified  Rules  for  Calculating  Lathe  Change 
Gears. — The  following  rules  for  calculating  change 
gears  for  the  lathe  have  been  published  by  Ma- 
chinery (Reference  Series  Book  No.  35,  Tables 


TRAINS  OF  MECHANISM  307 

and  Formulas  for  Shop  and  Draftingroom) ,  and 
are  here  given  because  of  their  concise  form  and 
simplicity. 

Rule  1. — To  find  the  "screw-cutting  constant*'  of 
a  lathe,  place  equal  gears  on  spindle  stud  and  lead- 
screw;  then  cut  a  thread  on  a  piece  of  work  in  the 
lathe.  The  number  of  threads  cut  with  equal 
gears  is  called  the  "  screw-cutting  constant "  of 
that  particular  lathe. 

Rule  2. — To  find  the  change  gears  used  in  simple 
gearing,  when  the  screw-cutting  constant  as  found 
by  Rule  1,  and  the  number  of  threads  per  inch  to 
be  cut  are  given,  place  the  screw-cutting  constant 
of  the  lathe  as  numerator  and  the  number  of  threads 
per  inch  to  be  cut  as  denominator  in  a  fraction,  and 
multiply  numerator  and  denominator  by  the  same 
number  until  a  new  fraction  is  obtained  represent- 
ing suitable  numbers  of  teeth  for  the  change  gears. 
In  the  new  fraction,  the  numerator  represents  the 
number  of  teeth  in  the  gear  on  the  spindle  stud, 
and  the  denominator,  the  number  of  teeth  in  the 
gear  on  the  lead-screw. 

Rule  3. — To  find  the  change  gears  used  in  com- 
pound gearing,  place  the  screw-cutting  constant  as 
found  from  Rule  1  as  numerator,  and  the  number 
of  threads  per  inch  to  be  cut  as  denominator  in  a 
fraction ;  divide  up  both  numerator  and  denomi- 
nator in  two  factors  each,  and  multiply  each  pair 
of  factors  (one  factor  in  the  numerator  and  one  in 
the  denominator  making  a  pair)  by  the  same  num- 
ber, until  new  fractions  are  obtained,  representing 
suitable  numbers  of  teeth  for  the  change  gears. 
The  gears  represented  by  the  numbers  in  the  new 


308         SELF-TAUGHT  MECHANICAL  DRAWING 

numerators  are  driving  gears,  and  those  in  the 
denominators  driven  gears. 

Two  examples,  showing  the  application  of  these 
rules,  will  be  given  in  the  following. 

Example  1. — Assume  that  20  threads  per  inch  are 
to  be  cut  in  a  lathe  having  a  "screw-cutting  con- 
stant," as  found  by  the  method  explained  in  Rule 
1,  equal  to  8.  The  numbers  of  teeth  in  the  avail- 
able change  gears  for  this  lathe  are  28,  32,  36,  40, 
44,  etc.,  increasing  by  4  up  to  96. 

By  applying  Rule  2,  we  have  then : 

S_        _8_X_4    =  32 
20  ==  20  X  4      80 

By  multiplying  both  numerator  and  denominator 
by  4  we  obtain  two  available  gears  having  32  and 
80  teeth.  The  32-tooth  gear  goes  on  the  spindle 
stud  and  the  80-tooth  gear  on  the  lead-screw.  It 
will  be  seen  that  if  we  had  multiplied  by  3  or  by  5 
instead  of  by  4,  we  would  not  have  obtained  avail- 
able gears  in  both  numerator  and  denominator,  as 
8X3  would  have  given  24  and  20  X  5  would  have 
given  100,  both  of  which  gears  are  not  in  our  given 
set  of  gears.  The  proper  number  by  which  to 
multiply  can  be  found  by  trial  only. 

Example  2. — Assume  that  27  threads  per  inch  are 
to  be  cut  on  the  same  lathe  as  assumed  in  Example  1. 

In  this  case  the  calculation  must  be  made  for 
compound  gearing,  as  so  fine  a  pitch  could  not  be 
cut  by  simple  gearing  in  this  lathe.  By  applying 
Rule  3  we  have : 

_8_       _2_>^4        (2  X  20)  X  (4  X  8)    =  40  X  32 
27  :  =   3  X  9       "(3~X  20)  X  (9  X  8)  =    60  X  72 


TRAINS  OF  MECHANISM 


309 


The  four  numbers  in  the  last  fraction  give  the 
numbers  of  teeth  in  the  required  gears.  The  gears 
in  the  numerator  (40  and  32)  are  the  driving  gears, 
and  those  in  the  denominator  (60  and  72)  are  the 
driven  gears. 

It  makes  no  difference  which  one  of  the  driving 
gears  is  placed  on  the  spindle  stud  or  which  one  of 
the  driven  gears  is  placed  on  the  lead-screw. 

Back-Gears.— Nearly  all  engine  lathes  and  many 
other  machine  tools  are  provided  with  a  set  of  re- 


c  = 


FIG.  212.— Principle  of  Back -Gearing. 

ducing  gears,  called  back-gears,  by  means  of  which 
double  the  range  of  speeds  that  can  be  obtained  by 
direct  driving  may  be  given  to  the  spindle  of  the 
machine.  Fig.  212  illustrates  such  a  set  of  gears, 
and  the  method  of  applying  them  to  the  machine. 
The  large  gear  A  is  fastened  to  the  spindle  of  the 
machine,  but  the  cone  pulley,  with  the  gear  B 
attached  to  it,  is  loose  on  the  spindle.  The  back- 


310         SELF-TAUGHT  MECHANICAL  DRAWING 

gear  shaft  with  gears  C  and  D  is  mounted  in 
brackets  on  the  back  side  of  the  head-stock,  and 
is  provided  with  eccentric  bearings,  by  means  of 
which  the  gears  on  it  can  be  thrown  into  or  out  of 
mesh  with  the  gears  on  the  head-stock  spindle. 
When  direct  driving  is  desired,  the  back-gears  are 
thrown  back,  out  of  the  way,  and  the  cone  pulley 
and  the  large  gear  are  clamped  together  by  means 
of  a  screw  pin  or  stud  passing  through  the  gear 
into  the  cone.  They  then  revolve  together  as  one 
piece. 

Let  us  assume  the  case  of  a  lathe  having  a  cone 
with  four  steps,  the  largest  step  being  6  inches  in 
diameter,  and  the  smallest  4  inches  in  diameter, 
with  the  intermediate  steps  in  proper  proportion. 
If  the  cone  pulley  on  the  countershaft  is  of  the 
same  size  as  the  one  on  the  spindle,  then,  if  the 
countershaft  runs  300  revolutions  per  minute,  direct 
driving  will  give  about  the  following  speeds  to  the 
spindle:  450,  345,  260  and  200.  Let  it  now  be 
required  to  find  the  sizes  of  gears  to  be  used  so 
that  with  the  back-gear  driving,  a  proportionately 
slower  rate  of  speeds  may  be  obtained.  We  may 
solve  the  problem  by  giving  to  the  gears  some 
arbitrary  sizes,  and  finding  what  speeds  such  sizes 
will  give,  and  then  modify  these  sizes  until  the 
required  speeds  are  obtained.  For  trial  purposes 
let  us  make  the  pitch  diameter  of  the  gear  A  the 
same  as  the  diameter  of  the  large  step  of  the  cone 
pulley,  or  6  inches,  and  the  pitch  diameter  of  the 
gear  B  the  same  as  the  diameter  of  the  small  step 
of  the  cone  pulley,  or  4  inches.  Arranging  driving 
and  driven  members  on  opposite  sides  of  a  vertical 


TRAINS  OF  MECHANISM  311 

line,  the  speed  of  the  first  driving  member  of  the 
train,  the  countershaft,  being  300,  the  required 
speed  of  the  last  member,  the  lathe  spindle,  being 
represented  by  x,  and  having  the  belt  on  the  largest 
step  of  the  countershaft  cone  so  as  to  obtain  the 
highest  speed  with  back-gears,  gives  an  arrange- 
ment of  the  case  as  below.  The  sizes  of  the  back- 
gears  are  the  same  as  those  on  the  lathe  spindle, 
the  gear  C  being  6  inches  in  pitch  diameter,  and 
the  gear  D  4  inches  in  pitch  diameter. 
Speed  of  countershaft  300 


Pulley  on  countershaft         6 
Gear  B  on  lathe  4 

Back-gear  D  4 


4  Pulley  on  lathe 
6  Back -gear  C 
6  Gear  A  on  lathe 


x  Speed  of  spindle 
28,800=  144s 


From  this  it  is  seen  that  with  the  sizes  of  the 
gears  as  above,  the  highest  speed  with  back-gears 
would  be  the  same  as  the  lowest  speed  without 
the  back-gears.  This,  of  course,  would  be  useless 
duplication  of  speeds. 

For  another  trial  we.  will  make  the  sizes  of  the 
gears  B  and  D  each  3J  inches  in  pitch  diameter. 
The  calculation  then  becomes: 


Speed  of  countershaft  300 

Pulley  on  countershaft  6 

Gear  B  on  lathe  3.5 

Back-gear  D  3,5 


4  Pulley  on  lathe 
6  Back-gear  C 
6  Gear  A  on  lathe 
x  Speed  of  spindle 


)  nearly. 


312         SELF-TAUGHT  MECHANICAL  DRAWING 

A  speed  of  153  revolutions  per  minute  for  the 
fastest  back-gear  speed  follows  quite  regularly  the 
series  of  speeds  which  the  direct  drive  gives. 

Instead  of  using  the  pitch  diameters  of  the  gears 
in  making  the  calculations  the  number  of  teeth 
which  the  gears  would  have,  the  pitch  being  first 
decided  on,  might  be  used.  In  this  manner  it  is 
possible  to  make  slight  changes  in  the  diameters  of 
the  gears  without  bringing  troublesome  fractions 
into  the  calculations. 

Many  lathes  and  other  machine  tools  have  trains 
of  mechanism  much  more  complicated  than  any 
here  shown,  but  the  method  of  procedure  here 
outlined  can  be  applied  to  all  of  them. 


CHAPTER  XX 

QUICK  RETURN  MOTIONS 

IN  a  large  class  of  machinery  the  work  is  done 
during  the  forward  motion  of  a  reciprocating  part; 
the  return  of  the  part  to  its  starting  point  is  then 
a  question  of  time.  The  quicker  the  part  can  be 
returned  to  its  starting  point,  the  more  efficient 
becomes  the  machine.  When  the  stroke  is  long,  as 
in  the  case  of  the  bed  of  an  iron  planer  for  large 
work,  this  rapid  return  motion  is  usually  obtained 
by  means  of  shifting  the  driving  belt  onto  a  return 
pulley  so  arranged  that  a  higher  ratio  of  speed  is 
procured;  but  in  other  cases,  where  the  recipro- 
cating motion  is  shorter,  and  the  stroke  is  actuated 
by  means  of  a  crank,  the  actuating  mechanism  is 
made  such  that  the  crank  gives  a  slow  forward 
and  a  quick  return  motion  to  the  reciprocating 
part.  Iron  planers  for  small  work,  shapers,  and 
the  like,  and  some  classes  of  engines  and  pumps, 
use  such  quick  return  motions.  Below  are  described 
the  principal  devices  used  for  such  purposes. 

Fig.  213  shows  a  method  of  securing  a  quick 
return  by  having  the  axis  of  the  crank  outside  of 
the  path  of  the  reciprocating  end  of  the  connecting- 
rod.  Let  A  be  a  crank,  the  crank-pin  of  which,  a, 
acting  upon  the  connecting-rod  B  represented  by 
the  heavy  line,  causes  the  block  b  to  move  back  and 

313 


314         SELF-TAUGHT   MECHANICAL  DRAWING 

forth  in  the  path  CD.  When  the  crank  is  in  the 
position  shown  the  block  is  at  the  extreme  left  of 
its  stroke,  the  connecting-rod  and  crank  being  in 
the  same  straight  line,  the  center  line  of  the  con- 
necting-rod coinciding  with  the  axis  of  the  crank. 
As  the  crank  swings  downward,  the  block  b  is 
driven  to  the  right;  but  an  examination . of  the 
illustration  will  show  that  the  crank  must  make 


FIG.  213.— Simple  Quick  Return  Motion. 

more  than  a  half  revolution  before  it  again  forms 
a  straight  line  with  the  connecting-rod,  which  it 
will  do  when  the  block  has  reached  its  extreme 
position  to  the  right.  As,  therefore,  the  block 
makes  its  movement  to  the  right  while  the  crank 
is  swinging  through  the  lower  angle  included  be- 
tween these  two  positions,  and  as  it  makes  its 
return  stroke  while  the  crank  is  swinging  through 
the  upper  angle  included  between  these  same  two 
positions,  the  time  of  the  forward  stroke  of  the 
block -will  be  to  the  time  of  its  return  stroke  as 
the  lower  angle  is  to  the  upper  angle. 


QUICK  RETURN  MOTIONS  315 

The  upper  angle  being  the  smaller  of  the  two, 
the  block  has  a  quick  return  motion.  To  secure 
ease  of  motion  to  the  block  as  it  starts  on  its  stroke 
to  the  right,  the  angle  abC,  the  angle  which  the 
connecting-rod  makes  with  the  path  of  the  block, 
should  not  be  more  than  about  45  degrees. 

To  design  a  quick  return  motion  of  this  type,  lay 
out  a  horizontal  line  ab,  Fig.  214,  and  on  it  mark 
off  cb  equal  to  the  required  length  of  stroke.  From 
c  draw  the  line  cd  of  indefinite  length  at  such  an 


fl  b 

FIG.  214.— Lay-out  of  Quick  Return  Motion  in  Fig.  213. 

obliquity  that  the  angle  acd  shall  not  be  more  than 
45  degrees.  From  b  draw  the  line  be  at  the  angle 
required  to  give  the  desired  quick  return.  The 
intersection  of  these  two  lines  at /will  be  the  axis 
of  the  crank.  The  length  bf  will  be  seen  by  re- 
ferring back  to  Fig.  213  to  be  equal  to  the  length 
of  the  crank  plus  the  length  of  the  connecting-rod. 
The  length  of  cf  will  be  seen  to  be  equal  to  the 
length  of  the  connecting-rod  minus  the  length  of 
the  crank.  If  in  a  given  case  the  length  cb  is 
made  12  inches,  and  cf  is  found  to  be  10  and  bf  21 
inches,  which  they  would  be  if  the  angles  were  as 


316    "     SELF-TAUGHT  MECHANICAL  DRAWING 

shown  in  Fig.  214,  then,  letting  x  represent  the 
length  of  the  connecting-rod  and  y  the  length  of 
the  crank,  we  would  have  x  +  y  =  21  inches,  and 
x  -  y  =  10  inches.  Adding  the  left-hand  and  the 
right-hand  members,  respectively,  of  these  two 
equations,  we  would  have  x  +  y  +  x-y  =  21  +  10 
=  31  inches.  As  +  y  -  y=  0  we  may  eliminate 
these  expressions,  and  the  equation  will  read  2x  = 
31  inches,  and  x,  the  length  of  the  connecting-rod, 
will  thus  be  15J  inches.  The  length  of  the  crank 
will  then  be  21  inches  (the  length  of  bf)  minus  15J 
inches,  or  5J  inches. 

It  will  be  seen  that  if  the  length  of  the  stroke  is 
made  variable  by  having  the  crank-pin,  a,  adjust- 
able to  different  positions  on  the  crank  A,  Fig.  213, 
the  difference  between  the  time  of  the  forward 
and  of  the  return  stroke  of  the  sliding  block  b  will 
be  lessened,  because  the  two  positions  which  it 
will  occupy  at.  the  extremes  of  its  stroke  will  be 
nearer  together,  and  the  lower  and  upper  angles 
which  the  crank  passes  through  in  giving  to  the 
block  its  forward  and  return  movements  will  be 
more  nearly  equal. 

Fig.  215  shows  a  quick  return  motion  device 
especially  adapted  to  cases  where  the  horizontal 
space  is  limited,  and  which  is  much  used  on  shapers. 
The  illustration  shows  a  shaper  in  outline.  The 
ram  of  the  shaper  is  given  its  forward  and  return 
motion  by  means  of  the  rocking  arm  A,  which 
swings  on  a  fulcrum  at  B.  The  rocking  arm  is 
given  its  motion  by  means  of  a  crank-pin  on  the 
disk  C,  the  pin  engaging  in  a  sliding  block  which 
travels  in  a  slot  in  the  arm  A. 


QUICK  RETURN  MOTIONS 


317 


Let  BC  and  BD,  Fig.  216,  represent  the  extreme 
positions  of  the  rocker  arm  A.  Draw  the  lines  OF 
and  OG  from  the  center  of  the  crank  disk  at  O  at 
right  angles  to  BC  and  BD.  It  is  evident  that  in 
order  that  the  crank,  on  its  upper  sweep,  shall 


FIG.  215.— Diagram  of  Quick  Return  Arrangement 
in  a  Shaper. 

move  the  rocker  arm  from  C  to  Z),  it  must  move 
through  the  arc  FAG,  while  to  return  the  arm 
from  D  to  C,  on  its  lower  sweep,  it  must  move  only 
through  the  lower  arc  FG.  The  time  of  the  return 
motion  will  therefore  be  to  the  time  of  the  forward 
motion  as  the  lower  arc  or  angle  FG  is  to  the  arc 


318         SELF-TAUGHT  MECHANICAL  DRAWING 

or  angle  FAG.  If  the  crank  is  shortened  so  as  to 
give  a  shorter  stroke  to- the  ram  of  the  shaper, 
then  the  rocker  arm  will  swing  through  a  smaller 
angle,  as  from  H  to  /,  and  lines  drawn  from  0  at 


FIG.  216.— Diagram  of  Speed  Ratios  in  Shaper  Motion. 

right  angles  to  HB  and  IB  will  be  more  nearly  in  a 
straight  line  than  OF  and  OG.  There  will,  there- 
fore, be  less  difference  between  the  time  of  forward 
and  return  motions  on  short  strokes  than  on  long 
ones. 


QUICK  RETURN  MOTIONS 


319 


The  Whitworth  Quick  Return  Device.— Let  A, 
Fig.  217,  be  a  slotted  arm  revolving  on  its  axis  at 
B.  Above  A  is  the  driving  crank  C,  having  a  pin 
engaging  in  the  slot  at  the  left  in  the  arm  A.  The 
slot  at  the  right  in  the  arm  A  is  provided  for  an 
adjustable  stud  which  drives  the  reciprocating 
parts,  through  the  medium  of  the  connecting-rod 


_D 


FIG.  217. -Whitworth  Quick  Return  Motion. 

D.  It  will  be  seen  that,  as  shown,  the  connecting- 
rod  is  at  the  extreme  right  of  its  motion,  forming 
as  it  does  a  straight  line  with  the  revolving  arm 
A,  which  latter  is  at  the  same  time  at  right  angles 
with  the  center  line  cd.  It  will  be  seen  that  in 
order  that  the  arm  A  may  move  through  half  a 
revolution  so  as  to  bring  the  connecting-rod  to  the 
extreme  left  of  its  motion,  it  will  be  necessary  for 
the  actuating  crank  C  to  revolve  either  through  the 


320         SELF-TAUGHT  MECHANICAL  DRAWING 

upper  angle  x  or  through  the  lower  angle  y,  so  as 
to  form  again  the  same  angle  with  the  center  line 
cd,  but  at  the  right  of  it,  as  it  is  now  shown  form- 
ing with  it  at  the  left.  The  forward  and  return 
motions  will,  therefore,  be  to  each  other  as  the 
angle  x  is  to  the  angle  y.  To  design  a  quick  return 
motion  of  this  type  it  is,  therefore,  necessary  to 
first  lay  out  the  angles  x  and  y  of  such  relative 
sizes  that  x  is  as  many  times  greater  than  y  as  the 
time  of  the  forward  motion  is  to  be  greater  than 
the  time  of  the  return  motion,  having  them,  of 
course,  central  on  the  line  cd.  The  distance  apart 
of  the  f  ulcrums  of  the  crank  C  and  of  the  revolving 
arm  A  will  be  partly  determined  by  the  sizes  of 
their  shafts.  The  location  of  the  crank-pin,  de- 
termining the  length  of  the  crank,  will  then  be  at 
the  intersection  of  the  horizontal  center  line  of  the 
revolving  arm  A  with  the  dividing  line  ef  between 
the  angles  x  and  y.  The  length  of  the  crank  must, 
of  course,  be  sufficient  so  that  the  crank  pin  will 
swing  under  the  hub  of  the  arm  A,  and  the  length 
of  the  crank-pin  slot  in  A  must  be.  sufficient  for 
the  motion  of  the  pin  relative  to  the  arm. 

It  will  be  noticed  that,  unlike  the  two  preceding 
quick  return  devices,  varying  the  stroke  of  the 
reciprocating  parts  does  not  alter  the  relative  time 
of  the  forward  and  return  motions ;  for  such  change 
does  not  affect  the  angles  x  and  y  upon  which  the 
time  of  the  forward  and  return  motions  depends. 
If,  however,  the  length  of  the  crank  C  is  varied, 
then  the  angles  x  and  y  are  altered,  and  the  time 
of  the  forward  and  return  motions  will  be  affected. 

It  will  be  seen  upon  examination  that  with  the 


QUICK  RETURN  MOTIONS 


321 


construction  shown  the  revolving  arm  A  must  be 
made  in  two  parts,  one  at  each  end  of  its  shaft,  in 
order  to  avoid  interference  of  the  parts  of  the 
mechanism  with  one  another  as  they  revolve.  This 
trouble  is  overcome  by  replacing  the  crank  C  with 
a  crank  disk  which  fits  over  and  revolves  upon  a 
fixed  stud  or  hub  large  enough  to  receive  the  stud 
at  B  upon  which  the  arm  A  revolves. 

The  Elliptic  Gear  Quick  Return. — If  two  ellipses 
of  equal  size,  Fig.  218,  having  foci  at  w  and  x  and 


FIG.  218.— Quick  Return  Motion  by  Means  of  Elliptic  Gears. 

at  y  and  z,  be  placed  in  contact  with  each  other 
with  their  long  diameters  forming  a  continuous 
straight  line  as  shown;  then  if  the  ellipses  are 
caused  to,  revolve  freely  upon  their  correspond- 
ing foci,  w  and  y,  they  will  roll  upon  each  other 
perfectly,  without  slipping.  From  the  nature 
of  an  ellipse  as  shown  by  its  construction  with  a 
thread  and  pencil  (see  Chapter  III,  Problem  13)  it 
will  be  seen  that  if  the  ellipse  at  the  left  were 
being  formed  in  this  manner  and  the  pencil  were 
at  D,  the  intersection  of  the  circumference  of  the 


322        SELF-TAUGHT  MECHANICAL  DRAWING 

ellipse  with  the  long  diameter,  the  length  of  the 
thread  would  be  equal  to  the  sum  of  the  distances 
wD  and  Dx.  But  the  distance  Dx  is  the  same  as 
the  distance  Dy\  therefore,  the  length  of  the  thread 
would  be  equal  to  the  distance  wy,  the  distance 
between  the  foci  upon  which  the  ellipses  are  re- 
volving. If,  now,  the  ellipses  are  revolved  until 
the  points  A  and  B,  vertically  over  the  foci  x  and 
y,  are  in  contact  with  each  other,  the  sum  of  the 
distances  wA  and  By  will  be  equal  to  the  distance 
between  the  foci  w  and  y,  for  their  sum  is  equal  to 
the  length  of  the  thread,  and  the  length  of  the 
thread  is  equal  to  wA  plus  Ax,  and  Ax  is  equal  to 
By,  as  points  A  and  B  are  both  vertically  over  the 
foci  of  the  ellipses.  In  a  similar  manner  any  pair 
of  points  may  be  selected  on  the  two  ellipses  equally 
distant  from  the  point  D.  The  distance  from  the 
point  on  the  ellipse  at  the  left,  to  the  focus  w,  will 
be  equal  to  the  length  of  the  thread  at  the  left  of 
the  pencil,  and  the  distance  from  the  point  on  the 
ellipse  at  the  right,  to  the  focus  y,  will  be  equal  to 
the  length  of  the  thread  at  the  right  of  the  pencil, 
and  their  sum  will  be  equal  to  the  distance  between 
the  foci  w  and  y.  This  distance  between  the  foci 
w  and  y  will  be  seen  on  further  examination  to  be 
equal  to  the  long  axis  of  the  ellipse.  This  property 
of  the  ellipse  has  been  taken  advantage  of  to  secure 
a  quick  return  motion  to  a  reciprocating  part  of  a 
machine.  If  in  Fig.  218  the  two  ellipses  represent 
the  pitch  lines  of  elliptic  gears;  with  the  gear  at 
the  left  as  the  driver  with  a  uniform  motion,  the 
one  at  the  right  will  have  an  ununiform  motion. 
If,  now,  a  crank  is  mounted  on  the  same  shaft  as 


QUICK  RETURN  MOTIONS  323 

the  driven  elliptic  gear,  the  crank  having  its  center 
line  at  right  angles  to  the  long  axis  of  the  ellipse, 
and  this  crank  actuates  a  sliding  block  back  and 
forth  in  the  direction  of  the  center  line  of  the  two 
gears,  then  this  block  will  have  a  slow  motion  in 
one  direction,  and  a  quick  motion  in  the  other 
direction.  If,  now,  the  gears  are  revolved  from  the 
position  in  which  they  are  shown  until  A  and  B 
are  in  contact,  the  gear  at  the  right  will  have  made 
a  quarter  of  a  revolution  and  the  sliding  block  will 
be  at  the  extreme  right  of  its  stroke;  but  while 
this  gear  has  made  a  quarter  of  a  revolution,  the 
driving  gear  has  revolved  through  the  angle  AwD 
only.  If,  now,  the  gear  at  the  right  is  revolved 
another  quarter  of  a  turn,  the  points  E  and  F  will 
be  in  contact,  and  the  crank  will  be  directed  ver- 
tically upward.  The  driving  gear  will,  however, 
have  revolved  through  the  angle  AwF.  The  forward 
and  return  motions  of  the  sliding  block  will,  there- 
fore, be  to  each  other  as  the  angle  AwF  is  to  the 
angle  AwD.  In  designing  a  pair  of  elliptic  gears, 
therefore,  the  first  thing  to  do  is  to  determine  the 
size  of  the  angle  Awx.  To  find  the  distance  be- 
tween the  foci  w  and  x  first  lay  out  on  a  large  scale 
a  triangle  similar  to  the  triangle  Awx.  Then  the 
sum  of  its  hypothenuse  and  the  perpendicular  will 
be  to  the  length  of  its  base  as  the  sum  of  wA  and 
Ax  (the  long  axis  of  the  ellipse)  is  to  wx,  the  dis- 
tance between  the  foci  of  the  ellipse.  The  length 
of  the  short  axis  may  then  be  found  by  reversing 
Problem  13,  Chapter  III.  The  problem  may  be 
solved  even  more  accurately  by  the  rules  given  for 
the  solution  of  right-angled  triangles.  The  length 


324         SELF-TAUGHT  MECHANICAL  DRAWING 


of  wA  will  be  to  Ax  as  1  is  to  the  sine  of  the  angle 
Awx.  Dividing  the  long  axis  of  the  ellipse  into 
two  parts  in  this  proportion  gives  the  length  of  wA 
and  Ax.  The  length  of  wx  will  then  be  equal  to 
the  length  of  Aw  multiplied  by  the  cosine  of  the 
angle  Awx.  Then  to  find  the  short  axis  of  the 
ellipse,  divide  the  distance  wx  into  two  equal  parts 
and  construct  the  triangle  wgh.  The  length  wh 
will  be  half  of  the  distance  between  the  foci,  and 
the  length  of  wg  will  be  half  of  the  long  axis.  The 
length  gh,  half  of  the  short  axis,  may  then  be  found. 
Calculations  made  in  this  manner  give  the  follow- 
ing proportions  to  ellipses  for  quick  return  ratios 
as  indicated  in  the  first  column : 


Ratio  of  Forward 
to  Return  Motion. 

Long  Axis. 

Short  Axis. 

Distance  Between 
Foci. 

2    to  1 

1.000 

0.963 

0.268 

2*  to  1 

1.000 

0.936 

0.351 

3    to  1 

1.000 

0.910 

0.414 

4    to  1 

1.000 

0.860 

0.509 

5    to  1 

1.000 

0.817 

0.577 

There  appear  to  be  two  difficulties  with  elliptic 
gearing.  The  first  is  that  if  a  high  quick  return 
ratio  is  attempted,  so  as  to  make  considerable  dif- 
ference between  the  long  and  the  short  axes,  the 
obliquity  of  the  action  of  the  teeth  upon  each 
other,  and  the  consequent  great  amount  of  friction 
between  the  teeth  as  they  come  together,  becomes 
so  great  as  to  be  troublesome.  This  may,  to  a  con- 
siderable extent  at  least,  be  overcome  by  using  a 
train  of  gears,  each  gear  but  slightly  elliptic,  in 
place  of  one  pair  of  decidedly  elliptic  form.  Thus 


QUICK  RETURN  MOTIONS  325 

a  train  of  three  gears  having  their  long  and  short 
axes  in  the  proportion  required  to  give  a  quick 
return  of  3  to  1,  with  one  pair  of  gears,  will  give 
a  quick  return  of  9  to  1.  If  three  gears  of  the  4  to 
1  proportion  are  usad,  a  quick  return  of  16  to  1 
will  result. 

The  second  difficulty  is  that  of  correctly  cutting 
the  teeth.  To  work  properly,  the  teeth  should  be 
cut  on  a  machine  having  a  special  elliptic  gear 
cutting  attachment,  otherwise  the  gears  are  likely 
to  be  expensive  and  unsatisfactory.  Such  an  ellip- 
tical gear  cutting  arrangement  is  described,  and 
the  subject  of  elliptic  gearing  is  quite  fully  dis- 
cussed, in  Grant's  treatise  on  gearing.  Not  being 
within  the  territory  of  this  elementary  treatise  on 
machine  design,  the  subject  cannot  here  be  dealt 
with  in  detail. 


INDEX 


Accelerated  motion  cams,  176 
Acceleration  of  falling  bodies, 

143 
Acme  standard  screw  thread, 

253 

Addendum  of  gear  teeth,  193 
Aluminum,  strength  of,  162 
Angle,  definition  of,  10 
Angle  of  cone  clutches,  271 
Angle,  to  bisect  an,  17 
Angles,  laying  out,  118 
Areas  of  plane  figures,  92 
A.  S.  M.  E.  standard  machine 

screws,  258 
Assembly  drawings,  52 


B 


Back  gears,  309 
Beams,  cross-sections  of,  156 
Beams,  strength  of,  159 
Belt  for  reversal  of  motion, 

crossed,  298 

Belting,  horse-power  of,  277 
Belting,  speed  of,  279 
Belting,  twisted  and  unusual 

cases  of,  282 
Belts,  276 
Belts,  endless,  278 
Belts,  laced,  278 
Belts,  width  and  thickness  of, 

277 
Bending,    shape  of  parts  to 

resist,  155 
Bending  strength  of  beams, 

159 


Bevel    gearing,    calculating, 

230 

Bevel  gears,  202 
Blue  printing,  78 
Bolt  heads,  table  of  United 

States  standard,  246 
Bolts,  studs  and  screws,  243 
Bolts  to  withstand  shock, 

248 

Brass,  strength  of  cast,  162 
Brass  wire,  strength  of,  158 
Broken  drawings  of  long  ob- 
jects, 73 


Cam  curve  for  harmonic  mo- 
tion, 181 

Cams,  comparison  between 
uniform  motion  and  accele- 
rated motion,  183 

Cams  for  high  velocities,  175 

Cams,  general  principles,  164 

Cams  with  grooved  edge,  172 

Cams  with  pivoted  follower, 
167 

Cams  with  positive  return, 
double,  173 

Cams  with  reciprocating  mo- 
tion, 171 

Cams  with  roller  follower, 
168 

Cams  with  straight  follower, 
165 

Cams  with  uniform  motion, 
165 

Cams  with  uniformly  accele- 
rated motion,  176 

Cap  screw  sizes,  248 


327 


328 


INDEX 


Case    for    drawing    instru- 
ments, 4 

Cast  iron,  strength  of,  157 
Castings,  stresses  in,  162 
Change  gears,  for  screw  cut- 
ting, 302 

Check  or  lock  nuts,  248 
Chord  of  circle,  definition  of, 

12 

Circle,  area   and   circumfer- 
ence of,  92 
Circle,  area  of,  83 
Circle,  circumference  of,  80 
Circle,  definition  of,  11 
Circle,  to   find   center  of  a, 

19 

Circles,  circumscribed  and  in- 
scribed, 20 

Circles,  concentric,  10 
Circles   in   isometric  projec- 
tion, 48 

Circular  pitch,  205 
Circular  sector,  area  of,  93 
Circular  segment,  area  of,  93 
Clamp  coupling,  262 
Clutches,  friction  cone,  269 
Clutches,  friction  disk,  266 
Clutches,  toothed,  265 
Compasses,  3 
Complement  angle,  definition 

of,  11 

Composition  of  forces,  120 
Compound  idler  gear,  301 
Compound  gearing  for  screw 

cutting,  305 
Compression     of     machine 

parts,  154 

Compressive  strength  of  ma- 
terials, 158 
Concentric  circles,  10 
Cone  and  cylinder  intersect- 
ing, 44 

Cone  clutches,  angle  of,  271 
Cone  clutches,  friction,  269 
Cone  pulleys,  239 
Cone  pulleys,  method  of  lay- 
ing out,  242 

Cone,    surface    development 
of  a,  40 


Copper,  strength  of  cast,  162 
Cosecant  of  an  angle,  102 
Cosine  of  an  angle,  101 
Cosines,  table  of,  105 
Cotangent  of  an  angle,  102 
Cotangents,  table  of,  107 
Coupling,  Hooke's,  263 
Couplings,  259 
Couplings,  clamp,  262 
Couplings,  flange,  260 
Crank  motion,  quick  return, 

313 

Cross-sectioning  device,  7 
Cross-sections  of  beams,  156 
Cube,  projections  of  a,  39 
Cube  root,  82 
Cube,  volume  of,  94 
Cutting  screw  threads,  gear- 
ing for,  302 

Cylinder  and  cone,  intersect- 
ing, 44 

Cylinder,  volume  of,  94 
Cylinders,  intersecting,  43 
Cycloid,  definition  of,  15 
Cycloid,  to  draw  a,  27 
Cycloidal  gear  teeth,  approx- 
imate shape  of,  209 


D 


Dedendum  of  gear  teeth,  193 
Definitions  of  terms,  10 
Degree,  definition  of,  96 
Detail  drawings,  53 
Diametral  pitch,  207 
Differential  pulleys,  134 
Disk  clutches,  friction,  266 
Dimensions  on  drawings,  56 
Double   cams    with   positive 

return,  173 

Drawings,  assembly,  52 
Drawing  board,  1 
Drawings,  classes  of  lines  on, 

55 

Drawings,  detail,  53 
Drawings,  dimensions  on,  56 
Drawing  instruments,  1 
Drawing  paper,  8 


INDEX 


329 


Drawing  pens,  the  use  of,  7 
Drawings,  sectional  views  on, 

66 
Drawings,  working,  50 


E 


Efficiency  of  screws,  253 
Elevation,  definition  of,  33 
Ellipse,  area  of,  95 
Ellipse,  definition  of,  14 
Ellipse,  to  draw  an,  21 
Elliptic    gear    quick    return 

motion,  321 
Elliptic  gear  return  motion, 

table  for  lay-out  of,  324 
Energy  and  work,  146 
Energy  of  fly-wheel,  290 
Engines,    horse-power     of 

steam,  81 

Epicycloid,  definition  of,  15 
Epicycloidal  gearing,  191 
Epicycloidal    and    involute 
systems  of  gears,  compari- 
son between,  199 
Erasing  shield,  9 


Factor  of  safety,  151 
Falling  bodies,  142 
Finishing  marks  on  drawings, 

63 

Flange  couplings,  260 
Foot-pound,  definition  of,  146 
Force  of  a  blow,  147 
Forces,  oblique,  124 
Forces,  opposing,  125 
Forces,  parallel,  123 
Forces,  resultant  of,  120 
Forces,  resolution  of,  123 
Formulas,  algebraic,  79 
Formulas,  transposition  of,  88 
Friction   cone   clutch,  horse- 
power of,  270 
Friction  cone  clutches,  269 
Friction   disk  clutch,    horse- 
power of,  267 


Friction  disk  clutches,  266 
Fulcrum,  definition  of,  126 
Fly-wheel,  energy  of,  290 
Fly-wheels     for    presses, 

punches,  etc.,  289 
Fly-wheel,  weight  of,  291 


Gear,  compound  idler,  301 

Gear,  influence  of  the  idler, 
299 

Gear  quick  return  motion, 
elliptic,  321 

Gear  teeth,  approximate 
shape  of,  209 

Gear  teeth,  laying  out  invo- 
lute, 210 

Gear  teeth,  Lewis'  formula 
for  strength  of,  218 

Gear  teeth,  pitch  of,  205 

Gear  teeth,  proportions  of, 
207 

Gear  teeth,  strength  of,  213 

Gear  teeth  systems,  compari- 
son between,  199 

Gear  tooth,  hunting,  209 

Gear  tooth  terms,  definitions 
of,  193 

Gear,  tumbler,  300 

Gearing,  back,  309 

Gearing,  calculating  bevel, 
230 

Gearing,  calculating  dimen- 
sions of,  222 

Gearing,  calculating  spur, 
222 

Gearing,  calculating  worm, 
234 

Gearing,  epicycloidal,  191 

Gearing  for  reversal  of  direc- 
tion of  motion,  299 

Gearing  for  screw  cutting, 
302 

Gearing,  general  principles 
of,  190 

Gearing,  worm,  204 

Gears,  bevel,  202 


330 


INDEX 


Gears,    interference    in    in- 
volute, 198 
Gears,  involute,  196 
Gears,  knuckle,  190 
Gears,  method  of  drawing,  68 
Gears,  proportions  of,  213 
Gears,  shrouded,  201 
Gears,  speed  ratio  of,  220 
Gears,  twenty   degree  invo- 
lute, 201 

Gears  with  radial  flanks,  195 
Gears  with  strengthened 

flanks,  195 

Geometrical  problems,  17 
Grooved  edge  cams,  172 
Guide  pulleys  for  belts,  285 


Instrument  case,  4 

Involute  and  epicycloidal  sys- 
tems of  gears,  comparison 
between,  199 

Involute,  definition  of,  15 

Involute  gears,  196 

Involute  gears,  interference 

.    in,  198 

Involute  gear  teeth,  laying 
out,  210 

Involute  gears,  twenty  de- 
gree, 201 

Involute  rack  teeth,  modified 
form  of,  197 

Involute,  to  draw  an,  27 

Iron  wire,  strength  of,  158 

Isometric  projection,  48 


Harmonic  motion  cam  curve, 

181 

Helix,  to  draw  a,  47 
Heptagon,  area  of,  94 
Hexagon,  area  of,  94 
Hexagon,  definition  of,  14 
Hexagon,  to  draw  a,  19 
Hoisting  pulleys,  132 
Hooke's  coupling  or  universal 

joint,  263 
Horse-power,  149 
Horse-power  of  belting,  277 
Horse-power  of  friction  cone 

clutch,  270 
Horse-power  of  friction  disk 

clutch,  267 

Horse-power  of  shafting,  274 
Horse-power    of   steam   en- 
gines, 81 

Hunting  tooth,  209 
Hypocycloid,  definition  of,  15 
Hypotenuse,  definition  of,  98 

I 

Idler  gear,  compound,  300 
Idler  gear,  influence  of  the, 

299 
Inclined  .plane,  136 


K 


Kirkaldy's  tests  on  strength 

of  materials,  157 
Knuckle  gears,  190 


Lathe  back  gearing,  309 

Lathe  change  gears,  302 

Lathe  change  gears,  simpli- 
fied rules  for  calculating, 
306 

Levers,  125 

Levers,  compound,  128 

Lewis'  formula  for  strength 
of  gear  teeth,  218 

Line,  definition  of,  10 

Line,  to  bisect  a,  17 

Lines  on  drawings,  classes  of, 
55 

Lock  or  check  nuts,  248 

M 

Machine  parts,  shape  of,  154 
Machine  screws,  257 
Machine   steel,  strength  of, 
158 


INDEX 


331 


Mechanics,  elements  of,  120 
Materials,  indicating,  72 
Mechanism,  trains  of,  297 
Metric   screw   thread,    form 

of,  256 

Minute,  definition  of,  97 
Moment,  twisting  or  torsion- 

al,  272 
Motion,    Newton's   laws   of, 

139 

N 

Newton's  laws  of  motion,  139 
Nuts,  check  or  lock,  248 
Nuts,  table  of  United  States 
standard,  246 


0 


Oblique-angled  triangles,  114 
Octagon,  area  of,  94 
Octagon,  definition  of,  14 
Octagon,  to  draw  an,  20 
Oldham's  coupling,  263 
Oscillation,  center  of,  141 


Paper,  drawing,  8 
Parallel  forces,  123 
Parabola,  definition  of,  15 
Parabola,  to  draw  a,  28 
Parallelogram,  area  of,  92 
Parallelogram,  definition  of, 

14 

Parallelogram  of  forces,  121 
Parallel  lines,  10 
Parenthesis  in  formulas,  85 
Pencils,  4 
Pendulum,  141 
Pens,  the  use  of  drawing,  7 
Pentagon,  area  of,  93 
Pentagon,  definition  of,  14 
Pentagon,  to  draw  a,  26 
Perpendicular  lines,  10 
Perpendicular  lines,  to  draw, 

18 


Pitch,  circular,  205 
Pitch  diameters,  table  of,  206 
Pitch,  diametral,  207 
Plane,  definition  of,  10 
Plane,  inclined,  136 
Point,  definition  of,  10 
Polygons,  definition  of,  14 
Positive  return  cams,  173 
Power  transmission,  screws 

for,  252 

Presses,  fly-wheels  for,  289 
Prism,  projections  of  a,  34 
Prism,  volume  of,  94 
Projection,  32 
Projection,  isometric,  48 
Pulley  diameters,  281 
Pulley   diameters,    to  calcu- 
late, 297 

Pulleys,  cone,  239 
Pulleys,  differential,  134 
Pulleys,  guide,  285 
Pulleys,  hoisting,  132 
Punches,  fly-wheels  for,  289 
Pyramid,     surface    develop- 
ment of  a,  41 
Pyramid,  volume  of,  94 


Q 


Quarter-turn  belting,  283 
Quick   return   device,  Whit- 
worth,  319 
Quick  return  motions,  313 


R 


Rack  teeth,  modified  form  of 

involute,  197 
Rack  with  epicycloidal  teeth, 

194 
Reciprocating   motion  cams, 

171 

Resolution  of  forces,  123 
Resultant  of  forces,  120 
Return  device,  Whitworth 

quick,  319 
Return  motion,  elliptic  gear 

quick,  321 


332 


INDEX 


Return  motions,  quick,  313 
Reversal  of  direction  of  mo- 
tion, to  secure,  298 
Right-angled  triangles,  97 


Safety,  factor  of,  151 

Scales,  2 

Screw  cutting,  gearing   for, 

302 

Screw,  differential,  138 
Screw,  in  mechanics,  138 
Screw  thread,   Acme  stand- 
ard, 253 

Screw  thread,  form  of  met- 
ric, 256 

Screw  thread,  sharp  V,  254 
Screw    thread,    Whitworth, 

255 

Screw  threads,  drawing,  74 
Screw     threads,    table    of 
United  States  standard,  246 
Screw  threads,  United  States 

standard,  245 
Screw  threads,  wrench  action 

on,  249 

Screws,  bolts  and  studs,  243 
Screws,  dimensioning,  62 
Screws,  efficiency  of,  253 
Screws  for  power   transmis- 
sion, 252 

Screws,  machine,  257 
Screws,  set,  256 
Screws,  square  threaded,  251 
Secant  of  an  angle,  102 
Second,  definition  of,  97 
Sections  on  drawings,  66 
Set-screws,  256 
Shade  lines,  77 
Shafting,  horse-power  of,  274 
Shafts,  272 

Shafts  at  right  angles,  belt- 
ing between,  283 
Shafts,   Thurston's   rule  for 

strength  of,  220 
Shapers,  quick    return    mo- 
tion for,  316 


Sharp  V-thread,  254 
Shearing  strength  of  mate- 
rials, 240 

Shearing  strength  of  shaft- 
ing, torsional,  273 
Shears,  fly-wheels  for  power, 

289 

Shrouded  gears,  201 
Sine  of  an  angle,  101 
Sines,  table  of,  104 
Solid,  definition  of,  10 
Speed  of  belting,  279 
Speed  ratio  of  gears,  220 
Speed   ratio   of  sprocket 

wheels,  189 
Speed,  to  secure  increase  of, 

297 
Sphere,  area  and  volume  of, 

94 
Spherical  sector,  volume  of, 

94 
Spherical    segment,    volume 

of,  95 

Spiral,  to  draw  a,  26 
Sprocket  wheels,  185 
Sprocket  wheels,  graphical 

method  of  laying  out,  187 
Sprocket  wheels,  speed  ratio 

of,  189 
Spur    gearing,     calculating, 

222 
Spur  gears,  method  of  draw- 

mg,  68 

Square  root,  82 
Square  threaded  screws,  251 
Steel  castings,  strength  of, 

157 
Steel,  strength  of  machine, 

158 
Steel,  strength  of  structural, 

162 

Steel  wire,  strength  of,  158 
Stepped  cone  pulleys,  239 
Strength  of  gear  teeth,  213 
Strength  of  gear   teeth, 

Lewis'  formula  for,  218 
Strength  of  materials,  151 
Strength  of: materials,  Kirk- 

aldy's  tests  on,  157 


INDEX 


333 


Strength  of  materials,  shear- 
ing, 260 

Strength  of  shafting,  tor- 
sional  shearing,  273 

Strength  of  shafts,  twisting, 
272 

Stresses  in  castings,  162 

Studs,  screws  and  bolts,  243 

Supplement  angle,  definition 
of,  11 

Surface,  definition  of,  10 


Tangent,  definition  of,  13 
Tangent  of  an  angle,  101 
Tangent  to  a  circle,  to  draw 

a,  19 

Tangents,  table  of,  106 
Tensile  strength  of  materials, 

158 

Tension  in  belts,  276 
Tension,  machine  parts  sub- 
jected to,  154 
Thickness  of  belts,  277 
Thread,   Acme    standard 

screw,  253 
Thread  cutting,  gearing  for, 

302 
Thread,  form  of  metric  screw, 

256 

Thread,  sharp  V,  254 
Thread,    Whitworth     screw, 

255 

Thread,  drawing  screw,  74 
Threads,  screws  with  square, 

251 
Threads,    United    States 

Standard  screw,  245 
Thurston's  rule  for  strength 

of  shafts,  220 
Toothed  clutches,  265 
Torsional  strength  of  shafts, 

272 
Trains  of  mechanism,  297 


Transposition  of  formulas,  88 
Triangle,  area  of,  91 
Triangles,  solution  of,  96 
Trigonometry,    elements   of, 

96 

Tumbler  gear,  300 
Twisting  strength  of  shafts, 

272 

U 

Uniform  motion  cams,  165 

Uniformly  accelerated  mo- 
tion cams,  176 

United  States  standard  screw 
thread,  245 

Universal  joint,  263 


V-Thread,  sharp,  254 
Vertex    of  angle,    definition 

of,  10 
Views  on  working  drawings, 

number  of,  50 
Volume  of  solids,  94 


w 

Weight  of  fly-wheel,  291 
Whitworth  quick  return  de- 
vice, 319 

Whitworth  screw  thread,  255 
Width  of  belts,  277 
Wire,  strength  of,  158 
Work  and  energy,  146 
Working  drawings,  50 
Worm  gearing,  204 
Worm  gearing,    calculating, 

234 
Wrench    action    on    screw 

threads,  249 

Wrought  iron,    strength  of, 
157 


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STANDARD 
PRACTICAL  AND 
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BOOKS 


PUBLISHED  AND  FOR  SALE  BY 

The  Norman  W,  Henley  Publishing  Go, 

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INDEX  OF  SUBJECTS 

Brazing  and  Soldering 3 

Cams ii 

Charts 3 

Chemistry 4 

Civil  Engineering 4 

C  oke 4 

Compressed  Air 4 

Concrete 5 

Dictionaries 5 

Dies— Metal  Work 6 

Drawing— Sketching  Paper 6 

Electricity 7 

Enameling 9 

Factory  Management,  etc 9 

Fuel 10 

Gas  Engines  and  Gas 10 

Gearing  and  Cams u 

Hydraulics n 

Ice  and  Refrigeration .'....- 1 1 

Inventions— Patents 12 

Lathe  Practice 12 

Liquid  Air 12 

Locomotive  Engineering 12 

Machine   Shop   Practice 14 

Manual  Training 17 

Marine  Engineering 17 

Metal  Work-Dies 6 

Mining 17 

Miscellaneous 18 

Patents  and  Inventions 12 

Pattern  Making 18 

Perfumery , 18 

Plumbing , 19 

Receipt  Book 24 

Refrigeration  and  Ice n 

Rubber 19 

Saws , '. 20 

Screw  Cutting 20 

Sheet  Metal  Work 20 

Soldering 3 

Steam  Engineering 20 

Steam  Heating  and  Ventilation 22 

Steam  Pipes ' 22 

Steel 22 

Watch  Making 23 

Wireless  Telephones 23 


Any  of  these  books  will  be  sent  prepaid  to   any   part  of 
the  world,  on  receipt  of  price. 

REMIT  by  Draft,  Postal  Money  Order,  Express  Money  Order 
or  by  Registered  Mail. 


GOOD,  USEFUL  BOOKS 


BRAZING    AND    SOLDERING 


BRAZING  AND  SOLDERING.  By  JAMES  F.  HOB  ART. 
The  only  book  that  shows  you  just  how  to  handle  any  job  of 
brazing  or  soldering  that  comes  along;  tells  you  what  mixture 
to  use,  how  to  make  a  furnace  if  you  need  one.  Full  of  kiaks. 
4th  edition.  25  cents 


CHARTS 


BATTLESHIP  CHART.  An  engraving  which  shows  the 
details  of  a  battleship  as  if  the  sides  were  of  glass  and  you  could 
see  all  the  interior.  The  finest  piece  of  work  that  has  ever  been 
done.  So  accurate  that  it  is  used  at  Annapolis  for  instruction 
purposes.  Shows  all  details  and  gives  correct  name  of  every 
part.  28  x  42  inches — plate  paper.  50  cents 

BOX  CAR  CHART.  A  chart  showing  the  anatomy  of  a  box 
car,  having  every  part  of  the  car  numbered  and  its  proper  name 
given  in  a  reference  list.  20  cents 


GONDOLA  CAR  CHART.  A  chart  showing  the  anatomy 
of  a  gondola  car,  having  every  part  of  the  car  numbered  and  its 
proper  reference  name  given  in  a  reference  list.  20  cents 


P ASSEN  GER  CAR  CHART.  A  chart  showing  the  anatomy 
of  a  passenger  car,  having  every  part  of  the  car  numbered  and  its 
proper  name  given  in  a  reference  list.  20  cents 

TRACTIVE  POWER  CHART.  A  chart  v/hereby  you  can 
find  the  tractive  power  or  drawbar  pull  of  any  locomotive, 
without  making  a  figure.  Shows  what  cylinders  are  equal,  how 
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WESTINGHOUSE  AIR-BRAKE  CHARTS.  Chart  I.— 
Shows  (in  colors)  the  most  modern  Westinghouse  High  Speed 
and  Signal  Equipment  used  on  Passenger  Engines,  Passenger 
Engine  Tenders,  and  Passenger  Cars.  Chart  II. — Shows  (in 
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and  Switch  Engines,  Freight  and  Switch  Engine  Tenders,  and 
Freight  Cars.  Price  for  the  set,  50  cents 


CHEMISTRY 

HENLEY'S      TWENTIETH      CENTURY      BOOK      OF 
RECEIPTS,    FORMULAS    AND    PROCESSES.      Edited    by 

GARDNER  D.  Hiscox.  The  most  valuable  Techno-chemical 
Receipt  Book  published,  including  over  10,000  selected  scientific 
chemical,  technological,  and  practical  receipts  and  processes. 
See  page  24  for  full  description  of  this  book.  $3.00 

CIVIL  ENGINEERING 


HENLEY'S  ENCYCLOPEDIA  OF  PRACTICAL  EN- 
GINEERING AND  ALLIED  TRADES.  Edited  by  JOSEPH 
G.  HORNER,  A.M.I.,  M.E.  This  set  of  five  volumes  contains 
about  2,500  pages  with  thousands  of  illustrations,  including  dia- 
grammatic and  sectional  drawings  with  full  explanatory  details. 
It  covers  the  entire  practice  of  Civil  and  Mechanical  Engineering. 
It  tells  you  all  you  want  to  know  about  engineering  and  tells  it 
so  simply,  so  clearly,  so  concisely  that  one  cannot  help  but 
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volumes. 

COKE 

COKE— MODERN  COKING  PRACTICE;  INCLUDING 
THE  ANALYSIS  OF  MATERIALS  AND  PRODUCTS. 

By  T.  H.  BYROM,  Fellow  of  the  Institute  of  Chemistry,  Fellow 
of  The  Chemical  Society,  etc.,  and  J.  E.  CHRISTOPHER,  Member 
of  the  Society  of  Chemical  Industry,  etc.  A  handbook  for 
those  engaged  in  Coke  manufacture  and  the  recovery  of  By- 
products. _  Fully  illustrated  with  folding  plates. 

The  subject  of  Coke  Manufacture  is  of  rapidly  increasing  in- 
terest and  significance,  embracing  as  it  does  the  recovery  of 
valuable  by-products  in  which  scientific  control  is  of  the  first 
importance.  It  has  been  the  aim  of  the  authors,  in  preparing 
this  book,  to  produce  one  which  shall  be  of  use  and  benefit  to 
those  who  are  associated  with,  or  interested  in,  the  modern  de- 
velopments of  the  industry. 

Contents:  Chap.  I.  Introductory.  Chap.  II.  General  Classi- 
fication of  Fuels.  Chap.  III.  Coal  Washing.  Chap.  IV.  The 
Sampling  and  Valuation  of  Coal,  Coke,  etc.  Chap.  V.  The 
Calorific  Power  of  Coal  and  Coke.  Chap.  VI.  Coke  Ovens. 
Chap.  VII.  Coke  Ovens,  continued.  Chap.  VIII.  Coke  Ovens, 
continued.  Chap.  IX.  Charging  and  Discharging  of  Coke  Ovens. 
Chap.  X.  Cooling  and  Condensing  Plant.  Ch'ap.  XI.  Gas  Ex- 
hausters. Chap.  XII.  Composition  and  Analysis  of  Ammoniacal 
Liquor.  Chap.  XIII.  Working  up  of  Ammoniacal  Liquor. 
Chap.  XIV.  Treatment  of  Waste  Gases  from  Sulphate  Plants. 
Chap.  XV.  Valuation  of  Ammonium  Sulphate.  Chap.  XVI. 
Direct  Recovery  of  Ammonia  from  Coke  Oven  Gases.  Chap. 
XVII.  Surplus  Gas  from  Coke  Oven.  Useful  Tables.  Very 
fully  illustrated.  83.50  net 

COMPRESSED    AIR 

COMPRESSED  AIR  IN  ALL  ITS  APPLICATIONS      By 

GARDNER  D.  Hiscox.  This  is  the  most  complete  book  on  the 
subject  of  Air  that  has  ever  been  issued,  and  its  thirty-five 
chapters  include  about  every  phase  of  the  subject  one  can  think 
of.  It  may  be  called  an  encyclopedia  of  compressed  air.  It  is 
written  by  an  expert,  who,  in  its  665  pages,  has  dealt  with  the 
subject  in  a  comprehensive  manner,  no  phase  of  it  being  omitted. 
Over  500  illustrations,  sth  Edition,  revised  and  enlarged. 
Cloth  bound;  $5.00,  Half  morocco,  *6.5O 


CONCRETE 

ORNAMENTAL  CONCRETE  WITHOUT  MOLDS,      By  A.  A. 

HOUGHTON.  The  process  for  making  ornamental  concrete  with- 
out molds,  has  long  been  held  as  a  secret  and  now,  for  the  first 
time,  this  process  is  given  to  the  public.  The  book  reveals  the 
secret  and  is  the  only  book  published  which  explains  a  simple, 
practical  method  whereby  the  concrete  worker  is  enabled,  by 
employing  wood  and  metal  templates  of  different  designs,  to 
mold  or  model  in  concrete  any  Cornice,  Archivolt,  Column, 
Pedestal,  Base  Cap,  Urn  or  Pier  in  a  monolithic  form — right 
upon  the  job.  These  may  be  molded  in  units  or  blocks,  and 
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is  fully  illustrated,  with  detailed  engravings.  83.00 

POPULAR  HAND  BOOK  FOR  CEMENT  AND  CON- 
CRETE USERS,  By  MYRON  H.  LEWIS,  C.E.  This  is  a  con- 
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manufacture  and  use  of  cement  in  all  classes  of  modern  works. 
The  author  has  brought  together  in  this  work,  all  the  salient 
matter  of  interest  to  the  user  of  concrete  and  its  many  diversified 
products.  The  matter  is  presented  in  logical  and  systematic 
order,  clearly  written,  fully  illustrated  and  free  from  involved 
mathematics.  Everything  of  value  to  the  concrete  user  is  given. 
Among  the  chapters  contained  in  the  book  are:  I.  Historical 
Development  of  the  Uses  of  Cement  and  Concrete.  II.  Glossary 
of  Terms  employed  in  Cement  and  Concrete  work.  III.  Kinds 
of  Cement  employed  in  Construction.  IV.  Limes,  Ordinary  and 
Hydraulic.  V.  Lime  Plasters.  VI.  Natural  Cements.  VII. 
Portland  Cements.  VIII.  Inspection  and  Testing.  IX.  Adul- 
teration; or  Foreign  Substances  in  Cement.  X.  Sand,  Gravel 
and  Broken  Stone.  XI.  Mortar.  XII.  Grout.  XIII.  Con- 
crete (Plain).  XIV.  Concrete  (Reinforced).  XV.  Methods 
and  Kinds  of  Reinforcements.  XVI.  Forms  for  Plain  and  Re- 
inforced Concrete.  XVII.  Concrete  Blocks.  XVIII.  Arti- 
ficial Stone.  XIX.  Concrete  Tiles.  XX.  Concrete  Pipes  and 
Conduits.  XXI.  Concrete  Piles.  XXII.  Concrete  Buildings. 
XXIII.  Concrete  in  Water  Works.  XXIV.  Concrete  in  Sewer 
Works.  XXV.  Concrete  in  Highway  Construction.  XXVI. 
Concrete  Retaining  Walls.  XXVII.  Concrete  Arches  and 
Abutments.  XXVIII.  Concrete  in  Subway  and  Tunnels. 
XXIX.  Concrete  in  Bridge  Work.  XXX.  Concrete  in  Docks 
and  Wharves.  XXXI.  Concrete  Construction  under  Water. 
XXXII.  Concrete  on  the  Farm.  XXXIII.  Concrete  Chimneys. 
XXXIV.  Concrete  for  Ornamentation.  XXXV.  Concrete 
Mausoleums  and  Miscellaneous  Uses.  XXXVI.  Inspection  for 
Concrete  Work.  XXX VII.  Waterproofing  Concrete  Work. 
XXXVIII.  Coloring  and  Painting  Concrete  Work.  XXXIX. 
Method  of  Finishing  Concrete  Surfaces.  XL.  Specifications  and 
Estimates  for  Concrete  Work.  $3.50 

DICTIONARIES 


STANDARD      ELECTRICAL      DICTIONARY.      By    T. 

O'CoNOR  SLOANE.  An  indispensable  work  to  all  interested  in 
electrical  science.  Suitable  alike  for  the  student  and  profession- 
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be  in  the  possession  of  all  who  desire  to  keep  abreast  with  the 
progress  of  this  branch  of  science.  Complete,  concise  and  con- 
venient. 682  pages — 393  illustrations.  83.00 


DIES— METAL   WORK 

DIES,  THEIR  CONSTRUCTION  AND  USE  FOR  THE 
MODERN  WORKING  OF  SHEET  METALS.  By  J.  V. 

WOODWORTH.  A  new  book  by  a  practical  man,  for  those  who 
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PUNCHES,  DIES  AND  TOOLS  FOR  MANUFACTUR- 
ING IN  PRESSES.  By  J.  V.  WOODWORTH.  An  encyclo- 
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ings. $4.00 

DRAWING— SKETCHING   PAPER 

LINEAR  PERSPECTIVE  SELF-TAUGHT.  By  HERMAN 
T.  C.  KRAUS.  This  work  gives  the  theory  and  practice  of  linear 
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SELF-TAUGHT  MECHANICAL  DRAWING  AND  ELE- 
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"Machinery."  A  practical  elementary  treatise  on  Mechanical 
Drawing  and  Machine  Design,  comprising  the  first  principles  of 
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A  NEW  SKETCHING  PAPER.  A  new  specially  ruled  paper 
to  enable  you  to  make  sketches  or  drawings  in  isometric  per- 
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ELECTRICITY 


ARITHMETIC  OF  ELECTRICITY.  By  Prof.  T.  O'CoNOR 
SLOANE.  A  practical  treatise  on  electrical  calculations  of  all 
kinds  reduced  to  a  series  of  rules,  all  of  the  simplest  forms,  and 
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pages.  $1.00 

COMMUTATOR  CONSTRUCTION.  By  WM.  BAXTER, 
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how  to  locate  troubles  and  how  to  remedy  them;  everyone  who 
fusses  with  dynamos  needs  this.  25  cents 

DYNAMO  BUILDING  FOR  AMATEURS,  OR  HOW  TO 
CONSTRUCT  A  FIFTY  WATT  DYNAMO.  By  ARTHUR 
J.  WEED,  Member  of  N.  Y.  Electrical  Society.  This  book  is  a 
practical  treatise  showing  in  detail  the  construction  of  a  small 
dynamo  or  motor,  the  entire  machine  work  of  which  can  be  done 
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Dimensioned  working  drawings  are  given  for  each  piece  of 
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The  book  is  illustrated  with  more  than  sixty  original  engrav- 
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Paper  50  cents     Cloth  81.00 

ELECTRIC  FURNACES  AND  THEIR  INDUSTRIAL 
APPLICATIONS.  By  J.WRIGHT.  This  is  a  book  which  will 
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who  desires  to  know  what  product  .can  be  manufactured  success- 
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who  merely  looks  into  the  subject  from  curiosity.  288  pages. 

$3.00 

ELECTRIC  LIGHTING  AND  HEATING  POCKET 
BOOK.  By  SYDNEY  F.  WALKER.  This  book  puts  in  conven- 
ient form  useful  information  regarding  the  apparatus  which  is 
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Tables  of  units  and  equivalents  are  included  and  useful  electrical 
laws  and  formulas  are  stated.  43  8  pages,  3  oo  engravings.  $3.00 

ELECTRIC  TOY  MAKING,  DYNAMO  BUILDING,  AND 
ELECTRIC  MOTOR  CONSTRUCTION.  This  work  treats 
of  the  making  at  home  of  electrical  toys,  electrical  apparatus, 
motors,  dynamos,  and  instruments  in  general,  and  is  designed  to 
bring  within  the  reach  of  young  and  old  the  manufacture  of  gen- 
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trated. $1.00 


ELECTRIC  WIRING,  DIAGRAMS  AND  SWITCH- 
BOARDS. By  NEWTON  HARRISON.  This  is  the  only  complete 
work  issued  snowing  and  telling  you  what  you  should  know 
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'ELECTRICIAN'S  HANDY  BOOK.  By  PROF.  T.  O'CpNOR 
SLOANE.  This  work  is  intended  for  the  practical  electrician, 
who  has  to  make  things  go.  The  entire  field  of  Electricity  is 
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thing is  to  the  point.  It  teaches  you  just  what  you  should 
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ELECTRICITY  IN  FACTORIES  AND  WORKSHOPS, 
ITS  COST  AND  CONVENIENCE.  By  ARTHUR  P.  HASLAM. 
A  practical  book  for  power  producers  and  power  users  showing 
what  a  convenience  the  electric  motor,  in  its  various  forms,  has 
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power.  312  pages.  Very  fully  illustrated.  $2.50 

ELECTRICITY  SIMPLIFIED.  By  PROF.  T.  O'CoNOR 
SLOANE.  The  object  of  "Electricity  Simplified"  is  to  make  the 
subject  as  plain  as  possible  and  to  show  what  the  modern  con- 
ception of  electricity  is;  to  show  how  two  plates  of  different 
metals  immersed  in  acid  can  send  a  message  around  the  globe; 
to  explain  how  a  bundle  of  copper  wire  rotated  by  a  steam  engine 
can  be  the  agent  in  lighting  our  streets,  to  tell  what  the  volt,  ohm 
and  ampere  are,  and  what  high  and  low  tension  mean;  and  to 
answer  the  questions  that  perpetually  arise  in  the  mind  in  this 
age  of  electricity.  172  pages.  Illustrated.  $1.00 

HOW    TO  BECOME    A  SUCCESSFUL  ELECTRICIAN. 

By  PROF.  T.  O'CoNOR  SLOANE.  An  interesting  book  from  cover 
to  cover.  Telling  in  simplest  language  the  surest  and  easiest  way 
to  become  a  successful  electrician.  The  studies  to  be  followed, 
methods  of  work,  field  of  operation  and  the  requirements  of  the 
successful  electrician  are  pointed  out  and  fully  explained. 
202  pages.  Illustrated.  $1.00 

MANAGEMENT  OF  DYNAMOS.  By  LUMMIS-PATER- 
SON.  A  handbook  of  theory  and  practice.  This  work  is  arranged 
in  three  parts.  The  first  part  covers  the  elementary  theory  of 
the  dynamo.  The  second  part,  the  construction  and  action  of 
the  different  classes  of  dynamos  in  common  use  are  described; 
while  the  third  part  relates  to  such  matters  as  affect  the  prac- 
tical management  and  working  of  dynamos  and  motors.  292 
pages,  117  illustrations.  $1.50 

STANDARD  ELECTRICAL  DICTIONARY.  By  Prof.  T. 
O'CoNOR  SLOANE.  A  practical  handbook  of  reference  contain- 
ing definitions  of  about  5,000  distinct  words,  terms  and  phrases. 
The  definitions  are  terse  and  concise  and  include  every  term 
used  in  electrical  science.  682  pages,  393  illustrations.  $3.00 

8 


SWITCHBOARDS.  By  WILLIAM  BAXTER,  JR.  This  book 
appeals  to  every  engineer  and  electrician  who  wants  to  know 
the  practical  side  of  things.  All  sorts  and  conditions  of  dynamos, 
connections  and-  circuits  are  shown  by  diagram  and  illustrate 
just  how  the  switchboard  should  be  connected.  Includes  direct 
and  alternating  current  boards,  also  those  for  arc  lighting,  in- 
candescent, and  power  circuits.  Special  treatment  on  high 
voltage  boards  for  power  transmission.  190  pages.  Illustrated. 

81.50 

TELEPHONE  CONSTRUCTION,  INSTALLATION, 
WIRING,  OPERATION  AND  MAINTENANCE.  By  W.  H. 

RADCLIFFE  and  H.  C.  CUSHING.  This  book  gives  the  principles 
of  construction  and  operation  of  both  the  Bell  and  Independent 
instruments;  approved  methods  of  installing  and  wiring  them; 
the  means  of  protecting  them  from  lightning  and  abnormal  cur- 
rents; their  connection  together  for  operation  as  series  or  bridg- 
ing stations;  and  rules  for  their  inspection  and  maintenance. 
Line  wiring  and  the  wiring  and  operation  of  special  telephone 
systems  are  also  treated.  180  pages,  125  illustrations.  81.00 

WIRING  A  HOUSE.  By  HERBERT  PRATT.  Shows  a  house 
already  built;  tells  just  how  to  start  about  wiring  it.  Where  to 
begin;  what  wire  to  use;  how  to  run  it  according  to  insurance 
rules,  in  fact  just  the  information  you  need.  Directions  apply 
equally  to  a  shop.  Fourth  edition.  35  cents 

WIRELESS  TELEPHONES  AND  HOW  THEY  WORK. 

By  JAMES  ERSKINE-MURRAY.  This  work  is  free  from  elaborate 
details  and  aims  at  giving  a  clear  survey  of  the  way  in  which 
Wireless  Telephones  work.  It  is  intended  for  amateur  workers 
and  for  those  whose  knowledge  of  Electricity  is  slight.  Chap- 
ters contained:  How  We  Hear — Historical — The  Conversion  of 
Sound  into  Electric  Waves  — Wireless  Transmission — The  Pro- 
duction of  Alternating  Currents  of  High  Frequency — How  the 
Electric  Waves  are  Radiated  and  Received — The  Receiving 
Instruments — Detectors — Achievements  and  Expectations — 
Glossary  of  Technical  Work.  Cloth.  81.00 


ENAMELING 


HENLEY'S  TWENTIETH  CENTURY  RECEIPT  BOOK. 

Edited  by  GARDNER  D.  Hiscox.  A  work  of  10,000  practical 
receipts,  including  enameling  receipts  for  hollow  ware,  for 
metals,  for  signs,  for  china  and  porcelain,  for  wood,  etc.  Thor- 
ough and  practical.  See  page  24  for  full  description  of  this  book. 

•3.00 

FACTORY  MANAGEMENT,  ETC. 


MODERN  MACHINE  SHOP  CONSTRUCTION,  EQUIP- 
MENT AND  MANAGEMENT.  By  O.  E.  PERRIGO,  M.E.  A 
work  designed  for  the  practical  and  every-day  use  of  the  Archi- 
tect who  designs,  the  Manufacturers  who  build,  the  Engineers 
who  plan  and.  equip,  the  Superintendents  who  organize  and 
direct,  and  for  the  information  of  every  stockholder,  director, 
officer,  accountant,  clerk,  superintendent,  foreman,  and  work- 
man of  the  modern  machine  shop  and  manufacturing  plant  of 
Industrial  America.  85.00 


FUEL 

COMBUSTION  OF  COAL  AND  THE  PREVENTION 
OF  SMOKE.  By  WM.  M.  BARR.  To  be  a  success  a  fireman 
must  be  "Light  on  Coal."  He  must  keep  his  fire  in  good  con- 
dition, and  prevent,  as  far  as  possible,  the  smoke  nuisance. 
To  do  this,  he  should  know  how  coal  burns,  how  smoke  is  formed 
and  the  proper  burning  of  fuel  to  obtain  the  best  results.  He 
can  learn  this,  and  more  too,  from  Barr's  "Combustion  of  Coal." 
It  is  an  absolute  authority  on  all  questions  relating  to  the  Firing 
of  a  Locomotive.  Nearly  350  pages,  fully  illustrated.  81.00 

SMOKE    PREVENTION  AND  FUEL  ECONOMY.      By 

BOOTH  and  KERSHAW.  As  the  title  indicates,  this  book  of  197 
pages  and  75  illustrations  deals  with  the  problem  of  complete 
combustion;  which  it  treats  from  the  chemical  and  mechanical 
standpoints,  besides  pointing  out  the  economical  and  humani- 
tarian aspects  of  the  question.  S2.5O 


GAS   ENGINES   AND   GAS 

CHEMISTRY  OF    GAS  MANUFACTURE.      By  H.    M. 

ROYLES.  A  practical  treatise  for  the  use  of  gas  engineers,  gas 
managers  and  students.  Including  among  its  contents — Prepa- 
rations of  Standard  Solutions,  Coal,  Furnaces,  Testing  and 
Regulation.  Products  of  Carbonization.  Analysis  of  Crude  Coal 
Gas.  Analysis  of  Lime.  Ammonia.  Analysis  of  Oxide  of  Iron. 
Naphthalene.  Analysis  of  Fire-Bricks  and  Fire-Clay .  Weldom 
and  Spent  Oxide.  Photometry  and  Gas  Testing.  Carbur- 
etted  Water  Gas.  Metropolis  Gas.  Miscellaneous  Extracts. 
Useful  Tables.  $4.50 

GAS  ENGINE  CONSTRUCTION,  Or  How  to  Build  a  Half- 
Horse-power  Gas  Engine.  By  PARSELL  and  WEED.  A  prac- 
tical treatise  describing  the  theory  and  principles  of  the  action  of 
gas  engines  of  various  types,  and  the  design  and  construction  of  a 
half-horse-power  gas  engine,  with  illustrations  of  the  work  in 
actual  progress,  together  with  dimensioned  working  drawings  giv- 
ing clearly  the  sizes  of  the  various  details.  300  pages.  $2.50 

GAS,  GASOLINE,  AND  OIL,  ENGINES.  By  GARDNER  D. 
Hiscox.  Just  issued,  i8th  revised  and  enlarged  edition.  Every 
user  of  a  gas  engine  needs  this  book.  Simple,  instructive,  and 
right  up-to-date.  The  only  complete  work  on  the  subject.  Tells 
all  about  the  running  and  management  of  gas,  gasoline  and  oil 
engines  as  designed  and  manufactured  in  the  United  States. 
Explosive  motors  for  stationary,  marine  and  vehicle  power  are 
fully  treated,  together  with  illustrations  of  their  parts  and  tabu- 
lated sizes,  also  their  care  and  running  are  included.  Electric 
Ignition  by  Induction  Coil  and  Jump  Sparks  are  fully  explained 
and  illustrated,  including  valuable  information  on  the  testing  for 
economy  and  power  and  the  erection  of  power  plants. 

The  special  information  on  PRODUCER  and  SUCTION  GASES  in- 
cluded cannot  fail  to  prove  of  value  to  all  interested  in  the  gen- 
eration of  producer  gas  and  its  utilization  in  gas  engines. 

The  rules  and  regulations  of  the  Board  of  Fire  Underwriters 
in  regard  to  the  installation  and  management  of  Gasoline  Motors 
is  given  in  full,  suggesting  the  safe  installation  of  explosive  motor 
power.  A  list  of  United  States  Patents  issued  on  Gas,  ^Gasoline 
and  Oil  Engines  and  their  adjuncts  from  1875  to  date  is  included. 
484  pages.  410  engravings.  S3. 50  net 


MODERN  GAS  ENGINES  AND  PRODUCER  GAS 
PLANTS.  By  R.  E.  MATHOT,  M.E.  A  practical  treatise  of 
320  pages,  fully  illustrated  by  175  detailed  illustrations,  setting 
forth  the  principles  of  gas  engines  and  producer  design,  the  selec- 
tion and  installation  of  an  engine,  conditions  of  perfect  opera- 
tion, producer-gas  engines  and  their  possibilities,  the  care  of  gas 
engines  and  producer-gas  plants,  with  a  chapter  on  volatile 
hydrocarbon  and  oil  engines.  This  book  has  been  endorsed  by 
Dugal  Clerk  as  a  most  useful  work  for  all  interested  in  Gas  Engine 
installation  and  Prodxicer  Gas.  82.50 


GEARING    AND   CAMS 


BEVEL  GEAR  TABLES.  By  D.  AG.  ENGSTROM.  No  one 
who  has  to  do  with  bevel  gears  in  any  way  should  be  without 
this  book.  The  designer  and  draftsman  will  find  it  a  great  con- 
venience, while  to  the  machinist  who  turns  up  the  blanks  or  cuts 
the  teeth,  it  is  invaluable,  as  all  needed  dimensions  are  given 
and  no  fancy  figuring  need  be  done.  81. OO 

CHANGE  GEAR  DEVICES.  By  OSCAR  E.  PERRIGO.  A 
book  for  every  designer,  draftsman  and  mechanic  who  is  inter- 
ested in  feed  changes  for  any  kind  of  machines.  This  shows  what 
has  been  done  and  how.  Gives  plans,  patents  and  all  information 
that  you  need.  Saves  hunting  through  patent  records  and  rein- 
venting old  ideas.  A  standard  work  of  reference.  81.00 

DRAFTING  OF  CAMS.  By  Louis  ROUILLION.  The 
laying  out  of  cams  is  a  serious  problem  unless  you  know  how  to 
go  at  it  right.  This  puts  you  on  the  right  road  for  practically 
any  kind  of  cam  you  are  likely  to  run  up  against.  25  cents 


HYDRAULICS 


HYDRAULIC  ENGINEERING.  By  GARDNER  D.  Hiscox. 
A  treatise  on  the  properties,  power,  and  resources  of  water  for  all 
purposes.  Including  the  measurement  of  streams;  the  flow  of 
water  in  pipes  or  conduits;  the  horse-power  of  falling  water; 
turbine  and  impact  water-wheels;  wave-motors,  centrifugal, 
reciprocating,  and  air-lift  pumps.  With  300  figures  and  dia- 
grams and  36  practical  tables.  320  pages.  84.00 


ICE    AND    REFRIGERATION 


POCKET  BOOK  OF  REFRIGERATION  AND  ICE  MAK- 
ING, By  A.  J.  WALLIS-TAYLOR.  This  is  one  of  the  latest  and 
most  comprehensive  reference  books  published  on  the  subject 
of  refrigeration  and  cold  storage.  It  explains  the  properties  and 
refrigerating  effect  of  the  different  fluids  in  use,  the  manage- 
ment of  refrigerating  machinery  and  the  construction  and  insula- 
tion of  cold  rooms  with  their  required  pipe  surface  for  different 
degrees  of  cold;  freezing  mixtures  and  non-freezing  brines, 
temperatures  of  cold  rooms  for  all  kinds  of  provisions,  cold 
storage  charges  for  all  classes  of  goods,  ice  making  and  storage  of 
ice,  data  and  memoranda  for  constant  reference  by  refrigerating 
engineers,  with  nearly  one  hundred  tables  containing  valuable 
references  to  every  fact  and  condition  required  in  the  installment 
and  operation  of  a  refrigerating  plant.  81.50 

II 


INVENTIONS— PATENTS 

INVENTOR'S  MANUAL,  HOW  TO  MAKE  A  PATENT 
PAY.  This  is  a  book  designed  as  a  guide  to  inventors  in  per- 
fecting their  inventions,  taking  out  their  patents,  and  disposing 
of  them.  It  is  not  in  any  sense  a  Patent  Solicitor's  Circular, 
nor  a  Patent  Broker's  Advertisement.  No  advertisements  of  any 
description  appear  in  the  work.  It  is  a  book  containing  a  quarter 
of  a  century's  experience  of  a  successful  inventor,  together  with 
notes  based  upon  the  experience  of  many  other  inventors.  $J  .00 

LATHE  PRACTICE 

MODERN  AMERICAN  LATHE  PRACTICE.  By  OSCAR 
E.  PERRIGO.  An  up-to-date  book  on  American  Lathe  Work, 
describing  and  illustrating  the  very  latest  practice  in  lathe  and 
boring-mill  operations,  as  well  as  the  construction  of  and  latest 
devel9pments  in  the  manufacture  of  these  important  classes  of 
machine  tools.  300  pages,  fully  illustrated.  83.50 

PRACTICAL  METAL  TURNING.  By  JOSEPH  G.  HORNER. 
A  work  of  404  pages,  fully  illustrated,  covering  in  a  comprehen- 
sive manner  the  modern  practice  of  machining  metal  parts  in 
the  lathe,  including  the  regular  engine  lathe,  its  essential  design, 
its  uses,  its  tools,  its  attachments,  and  the  manner  of  holding  the 
work  and  performing  the  operations.  The  modernized  engine 
lathe,  its  methods,  tools,  and  great  range  of  accurate  work.  The 
Turret  Lathe,  its  tools,  accessories  and  methods  of  performing 
its  functions.  Chapters  on  special  work,  grinding,  tool  holders, 
speeds,  feeds,  modern  tool  steels,  etc.,  etc.  $3.50 

TURNING  AND  BORING  TAPERS.  By  FRED  H.  COL- 
VIN.  There  are  two  ways  to  turn  tapers;  the  right  way  and 
one  other.  This  treatise  has  to  do  with  the  right  way;  it  tells 
you  how  to  start  the  work  properly,  how  to  set  the  lathe,  what 
tools  to  use  and  how  to  use  them,  and  forty^and  one  other  little 
things  that  you  should  know.  Fourth  edition.  25  cents 

LIQUID  AIR 

LIQUID  AIR  AND  THE  LIQUEFACTION  OF  GASES. 

By  T.  O'CoNOR  SLOANE.     Theory,  history,  biography,  practical 
applications,  manufacture.    365  pages.    Illustrated.  $2.00 

LOCOMOTIVE  ENGINEERING 


AIR-BRAKE  CATECHISM.  By  ROBERT  H.  BLACKALL. 
This  book  is  a  standard  text  book.  It  covers  the  Westinghouse 
Air-Brake  Equipment,  including  the  No.  5  and  the  No.  6  E  T 
Locomotive  Brake  Equipment;  the  K  (Quick-Service)  Triple 
Valve  for  Freight  Service;  and  the  Cross-Compound  Pump. 
The  operation  of  all  parts  of  the  apparatus  is  explained  in  detail, 
and  a  practical  way  of  finding  their  peculiarities  and  defects, 
with  a  proper  remedy,  is  given.  It  contains  2,000  questions  with 
their  answers,  which  will  enable  any  railroad  man  to  pass  any 
examination  on  the  subject  of  Air  Brakes.  Endorsed  and  used 
by  air-brake  instructors  and  examiners  on  nearly  every  rail- 
road in  the  United  States.  23d  Edition.  380  pages,  fully 
illustrated  with  folding  plates  and  diagrams.  $2.00 


AMERICAN  COMPOUND  LOCOMOTIVES.  By  FRED 
H.  COLVIN.  The  most  complete  book  on  compounds  published. 
Shows  all  types,  including  the  balanced  compound.  Makes 
everything  clear  by  many  illustrations,  and  shows  valve  setting, 
breakdowns  and  repairs.  142  pages.  $1.00 

APPLICATION  OF  HIGHLY  SUPERHEATED  STEAM 
TO  LOCOMOTIVES.  By  ROBERT  GARBE.  A  practical  book. 
Contains  special  chapters  on  Generation  of  Highly  Superheated 
Steam;  Superheated  Steam  and  the  Two-Cylinder  Simple 
Engine;  Compounding  and  Superheating;  Designs  of  Locomotive 
Superheaters;  Constructive  Details  of  Locomotives  using  Highly 
Superheated  Steam;  Experimental  and  Working  Results.  Illus- 
trated with  folding  plates  and  tables.  82.50 

COMBUSTION  OF  COAL  AND  THE  PREVENTION 
OF  SMOKE.  By  WM.  M.  BARR.  To  be  a  success  a  fireman 
must  be  "Light  on  Coal."  He  must  keep  his  fire  in  good  con- 
dition, and  prevent  as  far  as  possible,  the  smoke  nuisance. 
To  do  this,  he  should  know  how  coal  burns,  how  smoke  is  formed 
and  the  proper  burning  of  fuel  to  obtain  the  best  results.  He 
can  learn  this,  and  more  too,  from  Barr's  "Combination  of  Coal." 
It  is  an  absolute  authority  on  all  questions  relating  to  the  Firing 
of  a  Locomotive.  Nearly  350  pages,  fully  illustrated.  $1.00 

LINK  MOTIONS,  VALVES  AND  VALVE  SETTING.  By 

FRED  H.  COLVIN,  Associate  Editor  of  "American  Machinist. 
A  handy  book  that  clears  up  the  mysteries  of  valve  setting. 
Shows  the  different  valve  gears  in  use,  how  they  work,  and  why. 
Piston  and  slide  valves  of  different  types  are  illustrated  and 
explained.  A  book  that  every  railroad  man  in  the  motive- 
power  department  ought  to  have.  Fully  illustrated.  60  cents. 

LOCOMOTIVE  BOILER  CONSTRUCTION.  By  FRANK 
A.  KLEINHANS.  The  only  book  showing  how  locomotive 
boilers  are  built  in  modern  shops.  Shows  all  types  of  boilers 
used;  gives  details  of  construction;  practical  facts,  such  as 
life  of  riveting  punches  and  dies,  work  done  per  day,  allowance 
for  bending  and  flanging  sheets  and  other  data  that  means  dol- 
lars to  any  railroad  man.  421  pages,  334  illustrations.  Six 
folding  plates.  $3.00 

LOCOMOTIVE  BREAKDOWNS  AND  THEIR  REM- 
EDIES. By  GEO.  L.  FOWLER.  Revised  by  Wm.  W.  Wood, 
Air-Brake  Instructor.  Just  issued  1910  Revised  pocket  edition. 
It  is  put  of  the  question  to  try  and  tell  you  about  every  subject 
that  is  covered  in  this  pocket  edition  of  Locomotive  Breakdowns. 
Just  imagine  all  the  common  troubles  that  an  engineer  may  ex- 
pect to  happen  some  time,  and  then  add  all  of  the  unexpected 
ones,  troubles  that  could  occur,  but  that  you  had  never  thought 
about,  and  you  will  find  that  they  are  all  treated  with  the  very 
best  methods  of  repair.  Walschaert  Locomotive  Valve  Gear 
Troubles,  Electric  Headlight  Troubles,  as  well  as  Questions  and 
Answers  on  the  Air  Brake  are  all  included.  294  pages.  Fully 
illustrated.  $1.00 

LOCOMOTIVE  CATECHISM.  By  ROBERT  GRIMSHAW. 
27th  revised  and  enlarged  edition.  This  may  well  be  called  an 
encyclopedia  of  the  locomotive.  Contains  over  4,000  examina- 
tion questions  with  their  answers,  including  among  them  those 
asked  at  the  First,  Second  and  Third  year's  Examinations. 
825  pages,  437  illustrations  and  3  folding  plates.  $2.50 

13 


NEW  YORK  AIR-BRAKE  CATECHISM.  By  ROBERT 
H.  BLACKALL.  This  is  a  complete  treatise  on  the  New  York 
Air-Brake  and  Air-Signalling  Apparatus,  giving  a  detailed  de- 
scription of  all  the  parts,  their  operation,  troubles,  and  the 
methods  of  locating  and  remedying  the  same.  200  pages,  fully 
illustrated.  81.00 

POCKET-RAILROAD  DICTIONARY  AND  VADE  ME- 
CU!\I.  ^  By  FRED  H.  COLVIN,  Associate  Editor  "American 
Machinist."  Different  from  any  book  you  ever  saw.  Gives  clear 
and  concise  information  on  just  the  points  you  are  interested  in. 
It's  really  a  pocket  dictionary,  fully  illustrated,  and  so  arranged 
that  you  can  find  just  what  you  want  in  a  second  without  an 
index.  Whether  you  are  interested  in  Axles  or  Acetylene;  Com- 
pounds or  Counter  Balancing;  Rails  or  Reducing  Valves;  Tires 
or  Turntables,  you'll  find  them  in  this  little  book.  It's  very 
complete.  Flexible  cloth  cover,  200  pages.  81.00 

TRAIN  RULES  AND  DESPATCHING.     By  H.  A.  DALBY. 

Contains  the  standard  code  for  both  single  and  double  track  and 
explains  how  trains  are  handled  under  all  conditions.  Gives  all 
signals  in  colors,  is  illustrated  wherever  necessary,  and  the 
most  complete  book  in  print  on  this  important  subject.  Bound 
in  fine  seal  flexible  leather.  221  pages.  81.50 

WALSCHAERT     LOCOMOTIVE     VALVE     GEAR.     By 

WM.  W.  WOOD.  If  you  would  thoroughly  understand  the 
Walschaert  Valve  Gear,  you  should  possess  a  copy  of  this  book. 
The  author  divides  the  subject  into  four  divisions,  as  follows: 
I.  Analysis  of  the  gear.  II.  Designing  and  erecting  of  the  gear 
III.  Advantages  of  the  gear.  IV.  Questions  and  answers  re 
lating  to  the  Walschaert  Valve  Gear.  This  book  is  specially  valu- 
able to  those  preparing  for  promotion.  Nearly  200  pages.  $1.50 

WESTINGHOUSE  E  T  AIR-BRAKE  INSTRUCTION 
POCKET  BOOK  CATECHISM.  By  WM.  W.  WOOD,  Air-Brake 
Instructor.  A  practical  work  containing  examination  questions 
and  answers  on  the  E  T  Equipment.  Covering  what  the  E  T 
Brake  is.  How  it  should  be  operated.  What  to  do  when  de- 
fective. Not  a  question  can  be  asked  of  the  engineman  up  for 
promotion  on  either  the  No.  5  or  the  No.  6  E  T  equipment  that 
is  not  asked  and  answered  in  the  book.  If  you  want  to  thor- 
oughly understand  the  E  T  equipment  get  a  copy  of  this  book. 
It  covers  every  detail.  Makes  Air-Brake  troubles. and  examina- 
tions easy.  Fully  illustrated  with  colored  plates,  showing 
various  pressures.  82.00 

MACHINE   SHOP    PRACTICE 


AMERICAN  TOOL  MAKING  AND  INTERCHANGE- 
ABLE MANUFACTURING.  ^  By  J.  V.  WOODWORTH.  A 
practical  treatise  on  the  designing,  constructing,  use,  and  in- 
stallation of  tools,  jigs,  fixtures,  devices,  special  appliances, 
sheet-metal  working  processes,  automatic  mechanisms,  and 
labor-saving  contrivances;  together  with  their  use  in  the  lathe 
milling  machine,  turret  lathe,  screw  machine,  boring  mill,  power 
press,  drill,  sub  press,  drop  hammer,  etc.,  for  the  working  of 
metals,  the  production  of  interchangeable  machine  parts,  and 
the  manufacture  of  repetition  articles  of  metal.  560  pages, 
600  illustrations.  *4.0O 


HENLEY'S  ENCYCLOPEDIA  OF  PRACTICAL  EN- 
GINEERING AND  ALLIED  TRADES.  Edited  by  JOSEPH 
G.  HORNER.  A.M.I.Mech.I.  This  work  covers  the  entire  prac- 
tice of  Civil  and  Mechanical  Engineering.  The  best  known  ex- 
perts in  all  branches  of  engineering  have  contributed  to  these 
volumes.  The  Cyclopedia  is  admirably  well  adapted  to  the  needs 
of  the  beginner  and  the  self-taught  practical  man,  as  well  as  the 
mechanical  engineer,  designer,  draftsman,  shop  superintendent, 
foreman  and  machinist. 

•It  is  a  modern  treatise  in  five  volumes.  Handsomely  bound 
in  Half  Morocco,  each  volume  containing  nearly  500  pages,  with 
thousands  of  illustrations,  including  diagrammatic  and  sectional 
drawings  with  full  explanatory  details.  $35.00  for  the  com- 
plete set  of  five  volumes.  $6.00  per  volume,  when  ordered  singly. 

MACHINE  SHOP  ARITHMETIC.  By  COLVIN-CHENEY. 
Most  popular  book  for  shop  men.  Shows  how  all  shop  problems 
are  worked  out  and  "why."  Includes  change  gears  for  cutting 
any  threads;  drills,  taps,  shink  and  force  fits;  metric  system 
of  measurements  and  threads.  Used  by  all  classes  of  mechanics 
and  for  instruction  of  Y.  M.  C.  A.  and  other  schools.  Fifth 
edition.  131  pages.  50  cents 

MECHANICAL  MOVEMENTS,  POWERS,  AND  DE- 
VICES. By  GARDNER  D.  Hiscox.  This  is  a  collection  of  1890 
engravings  of  different  mechanical  motions  and  appliances,  ac- 
companied by  appropriate  text,  making  it  a  book  of  great  value 
to  the  inventor,  the  draftsman,  and  to  all  readers  with  mechanical 
tastes.  The  book  is  divided  into  eighteen  sections  or  chapters 
in  which  the  subject  matter  is  classified  under  the  following 
heads:  Mechanical  Powers,  Transmission  of  Power,  Measurement 
of  Power,  Steam  Power,  Air  Power  Appliances,  Electric  Power 
and  Construction,  Navigation  and  Roads,  Gearing,  Motion  and 
Devices,  Controlling  Motion,  Horological,  Mining,  Mill  and 
Factory  Appliances,  Construction  and  Devices,  Drafting  Devices, 
Miscellaneous  Devices,  etc.  nth  edition.  400  octavo  pages. 

$3.50 

MECHANICAL  APPLIANCES,  MECHANICAL  MOVE- 
MENTS AND  NOVELTIES  OF  CONSTRUCTION.  By 

GARDNER  D.  Hiscox.  This  is  a  supplementary  volume  to  the 
one  upon  mechanical  movements.  Unlike  the  first  volume, 
which  is  more  elementary  in  character,  this  volume  contains 
illustrations  and  descriptions  of  many  combinations  of  motions 
and  of  mechanical  devices  and  appliances  found  in  different  lines 
of  Machinery.  Each  device  being  shown  by  a  line  drawing  with 
a  description  showing  its  working  parts  and  the  method  of  opera- 
tion. From  the  multitude  of  devices  described,  and  illustrated, 
might  be  mentioned,  in  passing,  such  items  as  conveyors  and 
elevators,  Prony  brakes,  thermometers,  various  types  of  boilers, 
solar  engines,  oil-fuel  burners,  condensers,  evaporators,  Corliss 
and  other  valve  gears,  governors,  gas  engines,  water  motors  of 
various  descriptions,  air  ships,  motors  and  dynamos,  automobile 
and  motor  bicycles,  railway  block  signals,  car  couples,  link  and 
gear  motions,  ball  bearings,  breech  block  mechanism  for  heavy 
guns,  and  a  large  accumulation  of  others  of  equal  importance. 
1,000  specially  made  engravings.  396  octavo  pages.  $2.50 

These  two  volumes  sell  for  $2.50  each, 
but  when  the  twQ  volumes  are  ordered 

at  one  time  from  us,  we  send  them  prepaid  to  any  address  in  the 
world,  on  receipt  of  $4.00.  You  save  $i  by  ordering  the  two 
volumes  of  Mechanical  Movements  at  one  time. 

15 


MODERN  MACHINE  SHOP  CONSTRUCTION,  EQUIP- 
MENT AND  MANAGEMENT.  By  OSCAR  E.  PERRIGO. 
The  only  work  published  that  describes  the  Modern  Machine 
Shop  or  Manufacturing  Plant  from  the  time  the  grass  is  growing 
on  the  site  intended  for  it  until  the  finished  product  is  shipped. 
Just  the  book  needed  by  those  contemplating  the  erection  of 
modern  shop  buildings,  the  rebuilding  and  reorganization  of  old 
ones,  or  the  introduction  of  Modern  Shop  Methods,  Time  and 
Cost  Systems.  It  is  a  book  written  and  illustrated  by  a  prac- 
tical shop  man  for  practical  shop  men  who  are  too  busy  to  read 
theories  and  want  facts.  It  is  the  most  complete  all-around  book 
of  its  kind  ever  published.  400  large  quarto  pages,  225  original 
and  specially-made  illustrations.  $5.00 

MODERN  MACHINE  SHOP  TOOLS;  THEIR  CON- 
STRUCTION, OPERATION,  AND  MANIPULATION.  By 

W.  H.  VANDERVOORT.  A  work  of  555  pages  and  673  illustra- 
tions, describing  in  every  detail  the  construction,  operation,  and 
manipulation  of  both  Hand  and  Machine  Tools.  Includes 
chapters  on  filing,  fitting,  and  scraping  surfaces;  on  drills,  ream- 
ers, taps,  and  dies;  the  lathe  and  its  tools;  planers,  shapers, 
and  their  tools;  milling  machines  and  cutters;  gear  cutters  and 
gear  cutting;  drilling  machines  and  drill  work;  grinding  ma- 
chines and  their  work;  hardening  and  tempering;  gearing, 
belting  and  transmission  machinery;  useful  data  and  tables. 

$4.00 

THE  MODERN  MACHINIST.  By  JOHN  T.  USHER.  This 
book  might  be  called  a  compendium  of  shop  methods,  showing  a 
variety  of  special  tools  and  appliances  which  will  give  new  ideas 
to  many  mechanics  from  the  superintendent  down  to  the  man 
at  the  bench.  It  will  be  found  a  valuable  addition  to  any  machin- 
ist's library  and  should  be  consulted  whenever  a  new  or  difficult 
job  is  to  be  done,  whether  it  is  boring,  milling,  turning,  or  plan- 
ing, as  they  are  all  treated  in  a  practical  manner.  Fifth  edition. 
320  pages,  250  illustrations.  $2.50 

MODERN  MECHANISM.  Edited  by  PARK  BENJAMIN.  A 
practical  treatise  on  machines,  motors  and  the  transmission  of 
power,  being  a  complete  work  and  a  supplementary  volume  to 
Appleton's  Cyclopedia  of  Applied  Mechanics.  Deals  solely  with 
the  principal  and  most  useful  advances  of  the  past  few  years. 
959  pages  containing  over  1,000  illustrations;  bound  in  half 
morocco.  $4.00 

MODERN  MILLING  MACHINES :  THEIR  DESIGN, 
CONSTRUCTION  AND  OPERATION.  By  JOSEPH  G. 
HORNER.  This  book  describes  and  illustrates  the  Milling  Ma- 
chine and  its  work  in  such  a  plain,  clear,  and  forceful  manner, 
and  illustrates  the  subject  so  clearly  and  completely,  that  the 
up-to-date  machinist,  student,  or  mechanical  engineer  can  not 
afford  to  do  without  the  valuable  information  which  it  contains. 
It  describes  not  only  the  early  machines  of  this  class,  but  notes 
their  gradual  development  into  the  splendid  machines  of  the 
present  day,  giving  the  design  and  construction  of  the  various 
types,  forms,  and  special  features  produced  by  prominent 
manufacturers,  American  and  foreign.  304  pages,  300  illustra- 
tions. $4.00 

"  SHOP  KINKS."  By  ROBERT  GRIMSHAW.  This  shows 
special  methods  of  doing  work  of  various  kinds,  and  reducing 
cost  of  production.  Has  hints  and  kinks  from  some  of  the  largest 
shops  in  th'is  country  and  Europe.  You  are  almost  sure  to  find 
some  that  apply  to  your  work,  and  in  such  a  way  as  to  save  time 
and  trouble.  400  pages.  Fourth  edition.  $2.50 

16 


TOOLS  FOR  MACHINISTS  AND  WOOD  WORKERS, 
INCLUDING  INSTRUMENTS  OF  MEASUREMENT.  By 

JOSEPH  G.  HORNER.  A  practical  treatise  of  340  pages,  fully 
illustrated  and  comprising  a  general  description  and  classifica- 
tion of  cutting  tools  and  tool  angles,  allied  cutting  tools  for 
machinists  and  woodworkers;  shearing  tools;  scraping  tools; 
saws;  milling  cutters;  drilling  and  boring  tools;  taps  and  dies; 
punches  and  hammers;  and  the  hardening,  tempering  and 
grinding  of  these  tools.  Tools  for  measuring  and  testing  work, 
including  standards  of  measurement;  surface  plates;  levels; 
surface  gauges;  dividers;  calipers;  verniers;  micrometers; 
snap,  cylindrical  and  limit  gauges;  screw  thread,  wire  and 
reference  gauges,  indicators,  templets,  etc.  83. 50 

MANUAL  TRAINING 

ECONOMICS  OF  MANUAL,  TRAINING.  By  Louis 
ROUILLION.  The  only  book  that  gives  just  the  information 
needed  by  all  interested  in  manual  training,  regarding  buildings, 
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grades  of  the  work  from  the  Kindergarten  to  the  High  and  Nor- 
mal School.  Gives  itemized  lists  of  everything  needed  and  tells 
just  what  it  ought  to  cost.  Also  shows  where  to  buy  supplies. 

$1.50 

MARINE   ENGINEERING 

MARINE  ENGINES  AND  BOILERS,  THEIR  DESIGN 
AND  CONSTRUCTION.  By  DR.  G.  BAUER,  LESLIE  S. 
ROBERTSON,  and  S.  BRYAN  DONKIN.  This  work  is  clearly 
written,  thoroughly  systematic,  theoretically  sound;  while  the 
character  of  its  plans,  drawings,  tables,  and  statistics  is  without 
reproach.  The  illustrations  are  careful  reproductions  from 
actual  working  drawings,  with  some  well-executed  photographic 
views  of  completed  engines  and  boilers.  $9.00  net 

MINING 


*ORE  DEPOSITS  OF  SOUTH  AFRICA  WITH  A 
CHAPTER  ON  HINTS  TO  PROSPECTORS.  By  J.  P.  JOHN- 
SON. This  book  gives  a  condensed  account  of  the  ore-deposits 
at  present  known  in  South  Africa.  It  is  also  intended  as  a  guide 
to  the  prospector.  Only  an  elementary  knowledge  of  geology 
and  some  mining  experience  are  necessary  in  order  to  under- 
stand this  work.  With  these  qualifications,  it  will  materially 
assist  one  in  his  search  for  metalliferous  mineral  occurrences 
and,  so  far  as  simple  ores  are  concerned,  should  enable  one  to 
form  some  idea  of  the  possibilities  of  any  they  may  find. 

Among  the  chapters  given  are:  Titaniferous  and  Chromif- 
erous  Iron  Oxides — Nickel — Copper — Cobalt — Tin — Molyb- 
denum— Tungsten — Lead — Mercury — Antimony — I  r  o  n — Hints 
to  Prospectors.  Illustrated.  $2.00 

PRACTICAL  COAL  MINING.  By  T.  H.  COCKIN.  An  im- 
portant work,  containing  428  pages  and  213  illustrations,  com- 
plete with  practical  details,  which  will  intuitively  impart  to  the 
reader,  not  only  a  general  knowledge  of  the  principles  of  coal 
mining,  but  also  considerable  insight  into  allied  subjects.  The 
treatise  is  positively  up  to  date  in  every  instance,  and  should 
be  in  the  hands  of  every  colliery  engineer,  geologist,  mine 
operator,  superintendent,  foreman,  and  all  others  who  are  in- 
terested in  or  connected  with  the  industry.  $2.50 

17 


PHYSICS  AND  CHEMISTRY  OF  MINING.     By  T.  H. 

BYROM.  A  practical  work  for  the  use  of  all  preparing  for  ex- 
aminations in  mining  or  qualifying  for  colliery  managers'  cer- 
tificates. The  aim  of  the  author  in  this  excellent  book  is  to  place 
clearly  before  the  reader  useful  and  authoritative  data  which 
will  render  him  valuable  assistance  in  his  studies.  The  only  work 
of  its  kind  published.  The  information  incorporated  in  it  will 
prove  of  the  greatest  practical  utility  to  students,  mining  en- 
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terested in  the  present-day  treatment  of  mining  problems.  160 
pages.  Illustrated.  $3.00 

MISCELLANEOUS 


BRONZES.  Henley's  Twentieth  Century  Receipt  Book  con- 
tarns  many  practical  formulas  on  bronze  casting,  imitation 
bronze,  bronze  polishes,  renovation  of  bronze.  See  page  24  for 
full  description  of  this  book.  83.00 

EMINENT  ENGINEERS.  By  DWIGHT  GODDARD.  Every- 
one who  appreciates  the  effect  of  such  great  inventions  as  the 
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ments. 

Mr.  Goddard  has  selected  thirty-two  of  the  world's  engineers 
who  have  contributed  most  largely  to  the  advancement  of  our 
civilization  by  mechanical  means,  giving  only  such  facts  as  are  of 
general  interest  and  in  a  way  which  appeals  to  all,  whether 
mechanics  or  not.  280  pages,  35  illustrations.  $1.50 

LAWS  OF  BUSINESS,  By  THEOPHILUS  PARSONS,  LL.D. 
The  Best  Book  for  Business  Men  ever  Published.  Treats  clearly 
of  Contracts,  Sales,  Notes,  Bills  of  -Exchange,  Agency,  Agree- 
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Mortgages,  Liens,  Assignments,  Minors,  Married  Women,  Arbi- 
tration, Guardians,  Wills,  etc.  Three  Hundred  Approved  Forms 
are  given.  Every  Business  Man  should  have  a  copy  of  this  book 
for  ready  reference. .  The  book  is  bound  in  full  sheep,  and  Con- 
tains 864  Octavo  Pages.  Our  special  price.  $3.50 

PATTERN   MAKING 


PRACTICAL  PATTERN  MAKING.  By  F.  W.  BARROWS. 
This  is  a  very  complete  and  entirely  practical  treatise  on  the 
subject  of  pattern  making,  illustrating  pattern  work  in  wood  and 
metal.  From  its  pages  you  are  taught  just  what  you  should 
know  about  pattern  making.  It  contains  a  detailed  description 
of  the  materials  used  by  pattern  makers,  also  the  tools,  both 
those  for  hand  use,  and  the  more  interesting  machine  tools;  hav- 
ing complete  chapters  on  The  Band  Saw,  The  Buzz  Saw,  and  The 
Lathe.  Individual  patterns  of  many  different  kinds  are  fully 
illustrated  and  described,  and  the  mounting  of  metal  patterns  on 
plates  for  molding  machines  is  included.  $3.00 

PERFUMERY 


HENLEY'S  TWENTIETH  CENTURY  BOOK  OF  RE- 
CEIPTS, FORMULAS  AND  PROCESSES.  Edited  by  G.  D. 
Hiscox.  The  most  valuable  Techno-Chemical  Receipt  Book 
published.  Contains  over  10,000  practical  Receipts  many  of 
which  will  prove  of  special  value  to  the  perfumer,  a  mine  of  in- 
formation, up  to  date  in  every  respect.  Cloth,  $3.OO;  half 
morocco.  See  page  24  for  full  description  of  this  book.  $4.00 

18 


PERFUMES  AND  THEIR  PREPARATION.      By  G.  W. 

ASKINSON,  Perfumer.  A  comprehensive  treatise,  in  which 
there  has  been  nothing  omitted  that  could  be  of  value  to  the 
Perfumer.  Complete  directions  for  making  handkerchief  per- 
fumes, smelling-salts,  sachets,  fumigating  pastilles;  preparations 
for  the  care  of  the  skin,  the  mouth,  the  hair,  cosmetics,  hair  dyes 
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of  aromatic  substances;  their  nature,  tests  of  purity,  and 
wholesale  manufacture.  A  book  of  general,  as  well  as  profes- 
sional interest,  meeting  the  wants  not  only  of  the  druggist  and 
perfume  manufacturer,  but  also  of  the  general  public.  Third 
edition.  312  pages.  Illustrated.  $3.00 


PLUMBING 


MODERN  PLUMBING    ILLUSTRATED.       By     R      M. 

STARBUCK.  The  author  of  this  book,  Mr.  R.  M.  Starbuck,  is  one 
of  the  leading  authorities  on  plumbing  in  the  United  States.  The 
book  represents  the  highest  standard  of  plumbing  work.  It  has 
been  adopted  and  used  as  a  reference  book  by  the  United  States 
Government,  in  its  sanitary  work  in  Cuba,  Porto  Rico  and  the 
Philippines,  and  by  the  principal  Boards  of  Health  of  the  United 
States  and  Canada. 

It  gives  Connections,  Sizes  and  Working  Data  for  All  Fixtures 
and  Groups  of  Fixtures.  It  is  helpful  to  the  Master  Plumber  in 
Demonstrating  to  his  customers  and  in  figuring  work.  It  gives 
the  Mechanic  and  Student,  quick  and  easy  Access  to  the  best 
Modern  Plumbing  Practice.  Suggestions  for  Estimating  Plumb- 
ing Construction  are  contained  in  its  pages.  This  book  repre- 
sents, in  a  word,  the  latest  and  best  up-to-date  practice,  and 
should  be  in  the  hands  of  every  architect,  sanitary  engineer 
and  plumber  who  wishes  to  keep  himself  up  to  the  minute  on  this 
important  feature  of  construction.  400  octavo  pages,  fully 
illustrated  by  55  full- page  engravings.  84.00 


RUBBER 


HENLEY'S  TWENTIETH  CENTURY  BOOK  OF  RE- 
CEIPTS, FORMULAS  AND  PROCESSES.  Edited  by  GARD- 
NER D.  Hiscox.  Contains  upward  of  10,000  practical  receipts, 
......  .,  ,  ..«'...,..  „ 

S3.00 


including  among  them  formulas  on  artificial  rubber.    See  page 
24  for  full  description  of  this  book. 


RUBBER  HAND  STAMPS  AND  THE  MANIPULATION 
OF  INDIA  RUBBER.  By  T.  O'CoNOR  SLOANE.  This  book 
gives  full  details  on  all  points,  treating  in  a  concise  and  simple 
manner  the  elements  of  nearly  everything  it  is  necessary  to  under- 
stand for  a  commencement  in  any  branch  of  the  India  Rubber 
Manufacture.  The  making  of  all  kinds  of  Rubber  Hand  Stamps, 
Small  Articles  of  India  Rubber,  U.  S.  Government  Composi- 
tion, Dating  Hand  Stamps,  the  Manipulation  of  Sheet  Rubber, 
Toy  Balloons,  India  Rubber  Solutions,  Cements,  Blackings, 
Renovating  Varnish,  and  Treatment  for  India  Rubber  Shoes, 
etc.;  the  Hektograph  Stamp  Inks,  and  Miscellaneous  Notes, 
with  a  Short  Account  of  the  Discovery,  Collection,  and  Manufac- 
ture of  India  Rubber  are  set  forth  in  a  manner  designed  to  be 
readily  understood,  the  explanations  being  plain  and  simple. 
Second  edition.  144  pages.  Illustrated.  $1.00 

19 


SAWS 

SAW    FILING  AND    MANAGEMENT  OF  SAWS.      By 

ROBERT  GRIMSHAW.  A  practical  hand  book  on  filing,  gumming, 
swaging,  hammering,  and  the  brazing  of  band  saws,  the  speed, 
work,  and  power  to  run  circular  saws,  etc.  A  handy  book  for 
those  who  have  charge  of  saws,  or  for  those  mechanics  who  do 
their  own  filing,  as  it  deals  with  the  proper  shape  and  pitches  of 
saw  teeth  of  all  kinds  and  gives  many  useful  hints  and  rules  for 
gumming,  setting,  and  filing,  and  is  a  practical  aid  to  those  who 
use  saws  for  any  purpose.  New  edition,  revised  and  enlarged. 
Illustrated.  81.00 

SCREW  CUTTING 


THREADS  AND  THREAD  CUTTING.  By  COLVIN  and 
STABEL.  This  clears  up  many  of  the  mysteries  of  thread- 
cutting,  such  as  double  and  triple  threads,  internal  threads,  catch- 
ing threads,  use  of  hobs,  etc.  Contains  a  lot  of  useful  hints  and 
several  tables.  25  cents 

SHEET   METAL   WORK 

DIES,  THEIR  CONSTRUCTION  AND  USE  FOR  THE 
MODERN  WORKING  OF  SHEET  METALS.  By  J.  V. 

WOODWORTH.  A  new  book  by  a  practical  man,  for  those  who 
wish  to  know  the  latest  practice  in  the  working  of  sheet  metals. 
It  shows  how  dies  are  designed,  made  and  used,  and  those  who 
are  engaged  in  this  line  of  work  can  secure  many  valuable 
suggestions.  $3.00 

PUNCHES,  DIES  AND  TOOLS  FOR  MANUFACTUR- 
ING IN  PRESSES.  By  J.  V.  WOODWORTH.  A  work  of  5.00 
pages  and  illustrated  by  nearly  700  engravings,  being  an  en- 
cyclopedia of  die-making,  punch-making,  die  sinking,  sheet- 
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and  mechanical  combinations  for  punching,  cutting,  bending, 
forming,  piercing,  drawing,  compressing,  and  assembling  sheet- 
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STEAM   ENGINEERING 

AMERICAN  STATIONARY  ENGINEERING.      By  W. 

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ENGINE  RUNNER'S  CATECHISM.  By  RpBERT  GRIM- 
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ENGINE    TESTS  AND    BOILER  EFFICIENCIES.     By 

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MODERN  STEAM  ENGINEERING  IN  THEORY  AND 
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21 


STEAM  HEATING  AND  VENTILATION 

PRACTICAL  STEAM,  HOT-WATER  HEATING  AND 
VENTILATION.  By  A.  G.  KING.  This  book  is  the  standard 
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STEAM  PIPES 


STEAM  PIPES:  THEIR  DESIGN  AND  CONSTRUC- 
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tables  of  the  loss  of  heat  in  thermal  units  from  naked  and  felted 
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STEEL 


AMERICAN  STEEL  WORKER.  By  E.  R.  MARKHAM. 
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NER D.  Hiscox.  The  most  valuable  techno-chemical  Receipt 
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$3.00 

WATCH   MAKING 


HENLEY'S  TWENTIETH  CENTURY  BOOK  OF  RE- 
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formulas  including  many  watchmakers'  formulas.  $3.0O 

WATCHMAKER'S  HANDBOOK.  By  CLAUDIUS  SAUNIER. 
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WIRELESS  TELEPHONES 


WIRELESS  TELEPHONES   AND   HOW  THEY  WORK. 

By  JAMES  ERSKINE-MURRAY.  This  work  is  free  from  elaborate 
details  and  aims  at  giving  a  clear  survey  of  the  way  in  which 
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Glossary  of  Technical  Words.  Cloth.  $1.0O 


Henley's  Twentieth  Century 

Book  of 

Recipes,  Formulas 
and  Processes 

Edited  by  GARDNER  D.  HISCOX,  M.E. 
Price  $3. 00  Cloth  Binding  $4. 00  Half  Morocco  Binding 

Contains  over  10,000  Selected  Scientific,  Chemical, 

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Processes,  including  Hundreds  of 

So-Called  Trade  Secrets 

for  Every  Business 

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10781 


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